library(tidyverse)
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library(astsa)
library(vars)
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library(fpp)
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library(ggplot2)
#setwd("/Users/duleepa/Desktop/Time_Series/Project")
trip_data = readRDS("Monthly_Trip.rds")
head(trip_data)
ports <- trip_data %>%
  filter(Port_Name %in% c("Sault Sainte Marie","Detroit","Port Huron", "Buffalo Niagara Falls", "Alexandria Bay", "Ogdensburg", "Massena", "Champlain Rouses Point"))

dim(ports)
## [1] 29343     5
S_S_Marie <- ports %>%
  filter(Port_Name %in% c("Sault Sainte Marie")) %>%
  filter(!grepl("Passengers", Measure, ignore.case = TRUE)) %>%
  group_by(Year, Month) %>%
  summarise(Total = sum(Value))
## `summarise()` has grouped output by 'Year'. You can override using the
## `.groups` argument.
Detroit <- ports %>%
  filter(Port_Name %in% c("Detroit")) %>%
  filter(!grepl("Passengers", Measure, ignore.case = TRUE)) %>%
  group_by(Year, Month) %>%
  summarise(Total = sum(Value))
## `summarise()` has grouped output by 'Year'. You can override using the
## `.groups` argument.
Port_Huron <- ports %>%
  filter(Port_Name %in% c("Port Huron")) %>%
  filter(!grepl("Passengers", Measure, ignore.case = TRUE)) %>%
  group_by(Year, Month) %>%
  summarise(Total = sum(Value))
## `summarise()` has grouped output by 'Year'. You can override using the
## `.groups` argument.
B_Niagara <- ports %>%
  filter(Port_Name %in% c("Buffalo Niagara Falls")) %>%
  filter(!grepl("Passengers", Measure, ignore.case = TRUE)) %>%
  group_by(Year, Month) %>%
  summarise(Total = sum(Value))
## `summarise()` has grouped output by 'Year'. You can override using the
## `.groups` argument.
Alexandria <- ports %>%
  filter(Port_Name %in% c("Alexandria Bay")) %>%
  filter(!grepl("Passengers", Measure, ignore.case = TRUE)) %>%
  group_by(Year, Month) %>%
  summarise(Total = sum(Value))
## `summarise()` has grouped output by 'Year'. You can override using the
## `.groups` argument.
Ogdensburg <- ports %>%
  filter(Port_Name %in% c("Ogdensburg")) %>%
  filter(!grepl("Passengers", Measure, ignore.case = TRUE)) %>%
  group_by(Year, Month) %>%
  summarise(Total = sum(Value))
## `summarise()` has grouped output by 'Year'. You can override using the
## `.groups` argument.
Massena <- ports %>%
  filter(Port_Name %in% c("Massena")) %>%
  filter(!grepl("Passengers", Measure, ignore.case = TRUE)) %>%
  group_by(Year, Month) %>%
  summarise(Total = sum(Value))
## `summarise()` has grouped output by 'Year'. You can override using the
## `.groups` argument.
Champlain_Point <- ports %>%
  filter(Port_Name %in% c("Champlain Rouses Point")) %>%
  filter(!grepl("Passengers", Measure, ignore.case = TRUE)) %>%
  group_by(Year, Month) %>%
  summarise(Total = sum(Value))
## `summarise()` has grouped output by 'Year'. You can override using the
## `.groups` argument.
S_S_Marie_ts <- ts(S_S_Marie$Total, frequency = 12, start = c(1996, 4))
Detroit_ts <- ts(Detroit$Total, frequency = 12, start = c(1996, 4))
Port_Huron_ts <- ts(Port_Huron$Total, frequency = 12, start = c(1996, 4))

B_Niagara_ts <- ts(B_Niagara$Total, frequency = 12, start = c(1996, 4))
Alexandria_ts <- ts(Alexandria$Total, frequency = 12, start = c(1996, 4))
Ogdensburg_ts <- ts(Ogdensburg$Total, frequency = 12, start = c(1996, 4))
Massena_ts <- ts(Massena$Total, frequency = 12, start = c(1996, 4))
Champlain_Point_ts <- ts(Champlain_Point$Total, frequency = 12, start = c(1996, 4))
# Plot the first time series
tsplot(S_S_Marie_ts, col = "blue", ylim = c(0, max(c(S_S_Marie_ts, Detroit_ts, Port_Huron_ts))), ylab = "Volume")

# Add the second and third time series to the plot
lines(Detroit_ts, col = "red")
lines(Port_Huron_ts, col = "green")

# Add legend
legend("topright", legend = c("Sault Ste. Marie", "Detroit", "Port Huron"), col = c("blue", "red", "green"), lty = 1)

#Plot
tsplot(B_Niagara_ts, col = "blue", ylim = c(0, max(c(S_S_Marie_ts, Detroit_ts, Port_Huron_ts))), ylab = "Volume")

# Add lines with different colors for all ports
lines(Detroit_ts, col = "green")
lines(S_S_Marie_ts, col = "purple")
lines(Port_Huron_ts, col = "orange")
lines(Alexandria_ts, col = "red")
lines(Ogdensburg_ts, col = "yellow")
lines(Massena_ts, col = "cyan")
lines(Champlain_Point_ts, col = "magenta")

# Add legend
legend("topright", legend = c("Sault Ste. Marie", "Detroit", "Port Huron", "Alexandria", "Ogdensburg", "Massena", "Champlain Point"), 
       col = c("blue", "green", "purple", "orange", "red", "yellow", "cyan", "magenta"), lty = 1)

Detroit_modes <- ports %>%
  filter(Port_Name %in% c("Detroit")) %>%
  group_by(Year, Month, Measure) %>%
  summarise(Total = sum(Value)) %>%
  filter(!grepl("Passengers", Measure, ignore.case = TRUE))
## `summarise()` has grouped output by 'Year', 'Month'. You can override using the
## `.groups` argument.
#passenger
Detroit_personal_vehicles <- Detroit_modes %>%
  filter(Measure %in% "Personal Vehicles")

Detroit_buses <- Detroit_modes %>%
  filter(Measure %in% "Buses")

Detroit_trains <- Detroit_modes %>%
  filter(Measure %in% "Trains")

Detroit_personal_vehicles_ts <- ts(Detroit_personal_vehicles$Total, frequency = 12, start = c(1996, 4))
Detroit_buses_ts <- ts(Detroit_buses$Total, frequency = 12, start = c(1996, 4))
Detroit_trains_ts <- ts(Detroit_trains$Total, frequency = 12, start = c(1996, 4))

#freight
Detroit_Trucks <- Detroit_modes %>%
  filter(Measure %in% "Trucks")

Detroit_Truck_Loaded <- Detroit_modes %>%
  filter(Measure %in% "Truck Containers Loaded")

Detroit_Truck_Empty <- Detroit_modes %>%
  filter(Measure %in% "Truck Containers Empty")

Detroit_rail <- Detroit_modes %>%
  filter(grepl("Rail", Measure, ignore.case = TRUE))

Detroit_Trucks_ts <- ts(Detroit_Trucks$Total, frequency = 12, start = c(1996, 4))
Detroit_Truck_Loaded_ts <- ts(Detroit_Truck_Loaded$Total, frequency = 12, start = c(1996, 4))
Detroit_rail_ts <- ts(Detroit_rail$Total, frequency = 12, start = c(1996, 4))
# Plot the first time series
tsplot(Detroit_personal_vehicles_ts, col = "blue", ylim = c(0, max(c(Detroit_personal_vehicles_ts, Detroit_buses_ts, Detroit_trains_ts))), ylab = "Volume")

# Add the second and third time series to the plot
lines(Detroit_buses_ts, col = "red")
lines(Port_Huron_ts, col = "green")

# Add legend
legend("topright", legend = c("Detroit personal vehicles", "Detroit buses", "Detroit trains"), col = c("blue", "red", "green"), lty = 1)

# Plot the first time series
tsplot(Detroit_Trucks_ts, col = "blue", ylim = c(0, max(c(Detroit_Trucks_ts, Detroit_rail_ts))), ylab = "Volume")

# Add the second and third time series to the plot
lines(Detroit_rail_ts, col = "red")


# Add legend
legend("topright", legend = c("Detroit Trucks", "Detroit rail"), col = c("blue", "red", "green"), lty = 1)

multivariate_trip <- cbind(S_S_Marie, Detroit[,3], Port_Huron[,3], B_Niagara[,3], Alexandria[,3], Ogdensburg[,3], Massena[,3], Champlain_Point[,3])
## New names:
## • `Total` -> `Total...3`
## • `Total` -> `Total...4`
## • `Total` -> `Total...5`
## • `Total` -> `Total...6`
## • `Total` -> `Total...7`
## • `Total` -> `Total...8`
## • `Total` -> `Total...9`
## • `Total` -> `Total...10`
colnames(multivariate_trip)[3] <- "S_S_Marie"
colnames(multivariate_trip)[4] <- "Detroit"
colnames(multivariate_trip)[5] <- "Port_Huron"
colnames(multivariate_trip)[6] <- "B_Niagara"
colnames(multivariate_trip)[7] <- "Alexandria"
colnames(multivariate_trip)[8] <- "Ogdensburg"
colnames(multivariate_trip)[9] <- "Massena"
colnames(multivariate_trip)[10] <- "Champlain_Point"
multivariate_ts <- ts(multivariate_trip[3:10], frequency = 12, start = c(1996, 4))
class(multivariate_ts)
## [1] "mts"    "ts"     "matrix"
plot(multivariate_ts)

theme_set(theme_bw())
autoplot(multivariate_ts) +
  ggtitle("Port Time-Series") +
  ylab("Volume") +
  theme(plot.title = element_text(hjust = 0.5))

## Testing for stationarity
library(tidyverse)
library(lmtest)

adf_results <- apply(multivariate_ts, 2, function(x) adf.test(x)$p.value)
# p-values based on Port
(result_table <- tibble(Port = names(adf_results), p_value = adf_results))
library(fGarch)
## NOTE: Packages 'fBasics', 'timeDate', and 'timeSeries' are no longer
## attached to the search() path when 'fGarch' is attached.
## 
## If needed attach them yourself in your R script by e.g.,
##         require("timeSeries")
library(MTS)
## 
## Attaching package: 'MTS'
## The following object is masked from 'package:vars':
## 
##     VAR
# Differencing the whole mts
stnry = diffM(multivariate_ts) #difference operation on a vector of time series. Default order of differencing is 1.

# Retest
adf_stnry_results <- apply(stnry, 2, function(x) adf.test(x)$p.value)
## Warning in adf.test(x): p-value smaller than printed p-value

## Warning in adf.test(x): p-value smaller than printed p-value

## Warning in adf.test(x): p-value smaller than printed p-value

## Warning in adf.test(x): p-value smaller than printed p-value

## Warning in adf.test(x): p-value smaller than printed p-value

## Warning in adf.test(x): p-value smaller than printed p-value

## Warning in adf.test(x): p-value smaller than printed p-value

## Warning in adf.test(x): p-value smaller than printed p-value
# p-values based on Port
(result_table <- tibble(Port = names(adf_stnry_results), p_value = adf_stnry_results))
## VAR modeling
plot.ts(stnry, main = "Stationary Time Series")

autoplot(ts(stnry,
            start = c(1996,4),
            frequency = 12)) +
  ylab("Volume") +
  ggtitle("Stationary Time Series")

stnry_ts <- ts(stnry, frequency = 12, start = c(1996, 4))
VARselect(stnry_ts, lag.max=8, type="none")
## $selection
## AIC(n)  HQ(n)  SC(n) FPE(n) 
##      7      4      2      7 
## 
## $criteria
##                   1            2            3            4            5
## AIC(n) 1.547234e+02 1.538456e+02 1.530843e+02 1.526804e+02 1.525921e+02
## HQ(n)  1.550303e+02 1.544593e+02 1.540049e+02 1.539078e+02 1.541264e+02
## SC(n)  1.554912e+02 1.553812e+02 1.553877e+02 1.557515e+02 1.564311e+02
## FPE(n) 1.568793e+67 6.525513e+66 3.053103e+66 2.045282e+66 1.882973e+66
##                   6            7            8
## AIC(n) 1.525190e+02 1.522917e+02 1.523483e+02
## HQ(n)  1.543602e+02 1.544398e+02 1.548032e+02
## SC(n)  1.571258e+02 1.576663e+02 1.584907e+02
## FPE(n) 1.764932e+66 1.422735e+66 1.529442e+66
# Creating a VAR model with vars
var.a <- vars::VAR(stnry,
                   lag.max = 10, season = 12, #highest lag order for lag length selection according to the choosen ic
                   ic = "AIC", #information criterion
                   type = "none") #type of deterministic regressors to include
summary(var.a)
## 
## VAR Estimation Results:
## ========================= 
## Endogenous variables: S_S_Marie, Detroit, Port_Huron, B_Niagara, Alexandria, Ogdensburg, Massena, Champlain_Point 
## Deterministic variables: none 
## Sample size: 314 
## Log Likelihood: -26823.63 
## Roots of the characteristic polynomial:
## 0.8896 0.8896 0.8708 0.8708 0.8663 0.8663 0.8547 0.8547 0.8272 0.8272 0.826 0.826 0.8211 0.8211 0.8148 0.8148 0.7919 0.7919 0.7901 0.7901 0.7848 0.7848 0.7787 0.7787 0.7692 0.7692 0.7645 0.7645 0.7569 0.7569 0.7502 0.7502 0.7423 0.7423 0.7236 0.7236 0.6925 0.6925 0.6416 0.6416 0.6392 0.6392 0.5378 0.5378 0.5129 0.5129 0.359 0.1577
## Call:
## vars::VAR(y = stnry, type = "none", season = 12L, lag.max = 10, 
##     ic = "AIC")
## 
## 
## Estimation results for equation S_S_Marie: 
## ========================================== 
## S_S_Marie = S_S_Marie.l1 + Detroit.l1 + Port_Huron.l1 + B_Niagara.l1 + Alexandria.l1 + Ogdensburg.l1 + Massena.l1 + Champlain_Point.l1 + S_S_Marie.l2 + Detroit.l2 + Port_Huron.l2 + B_Niagara.l2 + Alexandria.l2 + Ogdensburg.l2 + Massena.l2 + Champlain_Point.l2 + S_S_Marie.l3 + Detroit.l3 + Port_Huron.l3 + B_Niagara.l3 + Alexandria.l3 + Ogdensburg.l3 + Massena.l3 + Champlain_Point.l3 + S_S_Marie.l4 + Detroit.l4 + Port_Huron.l4 + B_Niagara.l4 + Alexandria.l4 + Ogdensburg.l4 + Massena.l4 + Champlain_Point.l4 + S_S_Marie.l5 + Detroit.l5 + Port_Huron.l5 + B_Niagara.l5 + Alexandria.l5 + Ogdensburg.l5 + Massena.l5 + Champlain_Point.l5 + S_S_Marie.l6 + Detroit.l6 + Port_Huron.l6 + B_Niagara.l6 + Alexandria.l6 + Ogdensburg.l6 + Massena.l6 + Champlain_Point.l6 + sd1 + sd2 + sd3 + sd4 + sd5 + sd6 + sd7 + sd8 + sd9 + sd10 + sd11 
## 
##                      Estimate Std. Error t value Pr(>|t|)    
## S_S_Marie.l1       -5.755e-01  1.056e-01  -5.449 1.19e-07 ***
## Detroit.l1          2.490e-02  1.185e-02   2.102 0.036572 *  
## Port_Huron.l1       2.166e-02  3.916e-02   0.553 0.580733    
## B_Niagara.l1       -2.812e-02  1.643e-02  -1.711 0.088234 .  
## Alexandria.l1      -5.094e-02  8.084e-02  -0.630 0.529149    
## Ogdensburg.l1       5.165e-01  1.846e-01   2.798 0.005532 ** 
## Massena.l1          7.558e-02  7.429e-02   1.017 0.309965    
## Champlain_Point.l1 -7.886e-02  5.862e-02  -1.345 0.179707    
## S_S_Marie.l2       -4.003e-01  1.196e-01  -3.346 0.000942 ***
## Detroit.l2          1.962e-02  1.346e-02   1.458 0.146126    
## Port_Huron.l2       9.186e-02  4.585e-02   2.004 0.046175 *  
## B_Niagara.l2       -3.990e-02  1.984e-02  -2.011 0.045387 *  
## Alexandria.l2      -1.613e-01  9.916e-02  -1.626 0.105106    
## Ogdensburg.l2       2.735e-01  2.263e-01   1.209 0.227898    
## Massena.l2          8.369e-02  8.650e-02   0.968 0.334173    
## Champlain_Point.l2 -1.338e-01  6.979e-02  -1.918 0.056264 .  
## S_S_Marie.l3       -3.137e-01  1.255e-01  -2.500 0.013064 *  
## Detroit.l3          2.587e-02  1.416e-02   1.827 0.068797 .  
## Port_Huron.l3       7.184e-02  5.025e-02   1.430 0.154003    
## B_Niagara.l3       -5.052e-02  2.239e-02  -2.257 0.024867 *  
## Alexandria.l3      -2.903e-01  1.085e-01  -2.676 0.007944 ** 
## Ogdensburg.l3       2.521e-01  2.510e-01   1.004 0.316282    
## Massena.l3          6.892e-02  9.542e-02   0.722 0.470797    
## Champlain_Point.l3  2.798e-02  7.408e-02   0.378 0.705974    
## S_S_Marie.l4       -1.713e-01  1.234e-01  -1.388 0.166259    
## Detroit.l4          2.517e-02  1.419e-02   1.774 0.077334 .  
## Port_Huron.l4      -1.372e-03  5.002e-02  -0.027 0.978148    
## B_Niagara.l4       -3.678e-02  2.239e-02  -1.643 0.101658    
## Alexandria.l4      -2.538e-01  1.095e-01  -2.318 0.021265 *  
## Ogdensburg.l4       4.889e-01  2.500e-01   1.956 0.051552 .  
## Massena.l4          7.255e-02  9.471e-02   0.766 0.444334    
## Champlain_Point.l4 -1.283e-02  7.548e-02  -0.170 0.865194    
## S_S_Marie.l5       -1.729e-01  1.161e-01  -1.489 0.137849    
## Detroit.l5          3.841e-02  1.337e-02   2.873 0.004408 ** 
## Port_Huron.l5       3.758e-02  4.679e-02   0.803 0.422600    
## B_Niagara.l5       -2.247e-02  1.995e-02  -1.126 0.261049    
## Alexandria.l5      -1.311e-01  1.001e-01  -1.309 0.191691    
## Ogdensburg.l5       4.764e-02  2.267e-01   0.210 0.833740    
## Massena.l5          1.125e-01  8.522e-02   1.320 0.187928    
## Champlain_Point.l5  2.253e-02  7.222e-02   0.312 0.755274    
## S_S_Marie.l6       -2.481e-02  1.025e-01  -0.242 0.808938    
## Detroit.l6          1.538e-02  1.248e-02   1.232 0.219113    
## Port_Huron.l6       2.425e-02  3.990e-02   0.608 0.543853    
## B_Niagara.l6       -1.494e-02  1.627e-02  -0.918 0.359337    
## Alexandria.l6      -1.202e-01  8.127e-02  -1.479 0.140494    
## Ogdensburg.l6       5.903e-02  1.873e-01   0.315 0.752948    
## Massena.l6          6.195e-02  7.266e-02   0.853 0.394720    
## Champlain_Point.l6 -3.186e-02  6.051e-02  -0.526 0.599039    
## sd1                 3.640e+04  4.280e+03   8.503 1.58e-15 ***
## sd2                 4.271e+03  5.316e+03   0.803 0.422463    
## sd3                 8.025e+03  4.743e+03   1.692 0.091888 .  
## sd4                 9.510e+03  5.596e+03   1.699 0.090491 .  
## sd5                 3.327e+04  4.727e+03   7.038 1.80e-11 ***
## sd6                 7.828e+03  5.553e+03   1.410 0.159871    
## sd7                 2.462e+04  5.334e+03   4.615 6.22e-06 ***
## sd8                 1.537e+04  4.568e+03   3.364 0.000887 ***
## sd9                 5.955e+03  4.979e+03   1.196 0.232824    
## sd10                1.699e+04  5.966e+03   2.848 0.004760 ** 
## sd11                2.399e+04  5.739e+03   4.179 4.02e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 
## Residual standard error: 8614 on 255 degrees of freedom
## Multiple R-Squared: 0.8494,  Adjusted R-squared: 0.8146 
## F-statistic: 24.38 on 59 and 255 DF,  p-value: < 2.2e-16 
## 
## 
## Estimation results for equation Detroit: 
## ======================================== 
## Detroit = S_S_Marie.l1 + Detroit.l1 + Port_Huron.l1 + B_Niagara.l1 + Alexandria.l1 + Ogdensburg.l1 + Massena.l1 + Champlain_Point.l1 + S_S_Marie.l2 + Detroit.l2 + Port_Huron.l2 + B_Niagara.l2 + Alexandria.l2 + Ogdensburg.l2 + Massena.l2 + Champlain_Point.l2 + S_S_Marie.l3 + Detroit.l3 + Port_Huron.l3 + B_Niagara.l3 + Alexandria.l3 + Ogdensburg.l3 + Massena.l3 + Champlain_Point.l3 + S_S_Marie.l4 + Detroit.l4 + Port_Huron.l4 + B_Niagara.l4 + Alexandria.l4 + Ogdensburg.l4 + Massena.l4 + Champlain_Point.l4 + S_S_Marie.l5 + Detroit.l5 + Port_Huron.l5 + B_Niagara.l5 + Alexandria.l5 + Ogdensburg.l5 + Massena.l5 + Champlain_Point.l5 + S_S_Marie.l6 + Detroit.l6 + Port_Huron.l6 + B_Niagara.l6 + Alexandria.l6 + Ogdensburg.l6 + Massena.l6 + Champlain_Point.l6 + sd1 + sd2 + sd3 + sd4 + sd5 + sd6 + sd7 + sd8 + sd9 + sd10 + sd11 
## 
##                      Estimate Std. Error t value Pr(>|t|)    
## S_S_Marie.l1        3.206e-01  7.392e-01   0.434  0.66484    
## Detroit.l1         -4.236e-01  8.294e-02  -5.107 6.42e-07 ***
## Port_Huron.l1       4.974e-02  2.741e-01   0.181  0.85618    
## B_Niagara.l1       -3.385e-02  1.150e-01  -0.294  0.76879    
## Alexandria.l1      -6.091e-01  5.658e-01  -1.077  0.28271    
## Ogdensburg.l1       9.913e-01  1.292e+00   0.767  0.44365    
## Massena.l1          6.027e-01  5.201e-01   1.159  0.24754    
## Champlain_Point.l1 -1.090e-01  4.103e-01  -0.266  0.79066    
## S_S_Marie.l2        2.788e-02  8.373e-01   0.033  0.97346    
## Detroit.l2         -2.080e-01  9.419e-02  -2.208  0.02812 *  
## Port_Huron.l2      -2.623e-03  3.209e-01  -0.008  0.99349    
## B_Niagara.l2       -8.478e-02  1.389e-01  -0.610  0.54211    
## Alexandria.l2      -8.077e-01  6.941e-01  -1.164  0.24563    
## Ogdensburg.l2      -1.248e+00  1.584e+00  -0.788  0.43151    
## Massena.l2          3.784e-01  6.055e-01   0.625  0.53260    
## Champlain_Point.l2 -4.409e-02  4.885e-01  -0.090  0.92816    
## S_S_Marie.l3       -7.068e-01  8.786e-01  -0.805  0.42183    
## Detroit.l3         -1.908e-01  9.911e-02  -1.925  0.05535 .  
## Port_Huron.l3      -3.833e-02  3.517e-01  -0.109  0.91331    
## B_Niagara.l3        1.654e-02  1.567e-01   0.106  0.91600    
## Alexandria.l3      -1.783e+00  7.594e-01  -2.347  0.01967 *  
## Ogdensburg.l3      -1.654e-01  1.757e+00  -0.094  0.92509    
## Massena.l3          2.441e-01  6.680e-01   0.365  0.71512    
## Champlain_Point.l3  3.970e-01  5.185e-01   0.766  0.44455    
## S_S_Marie.l4        2.801e-01  8.635e-01   0.324  0.74594    
## Detroit.l4          2.505e-02  9.936e-02   0.252  0.80112    
## Port_Huron.l4       2.895e-02  3.502e-01   0.083  0.93416    
## B_Niagara.l4       -1.225e-01  1.567e-01  -0.782  0.43506    
## Alexandria.l4      -1.688e+00  7.665e-01  -2.202  0.02854 *  
## Ogdensburg.l4       1.909e+00  1.750e+00   1.091  0.27621    
## Massena.l4          3.694e-01  6.629e-01   0.557  0.57784    
## Champlain_Point.l4 -3.904e-01  5.284e-01  -0.739  0.46068    
## S_S_Marie.l5        2.628e-01  8.129e-01   0.323  0.74672    
## Detroit.l5          2.404e-01  9.359e-02   2.569  0.01078 *  
## Port_Huron.l5      -2.134e-03  3.275e-01  -0.007  0.99481    
## B_Niagara.l5       -1.834e-01  1.396e-01  -1.314  0.19012    
## Alexandria.l5      -8.781e-01  7.008e-01  -1.253  0.21138    
## Ogdensburg.l5      -1.632e+00  1.587e+00  -1.028  0.30484    
## Massena.l5          1.727e-01  5.965e-01   0.290  0.77242    
## Champlain_Point.l5  2.677e-01  5.055e-01   0.530  0.59684    
## S_S_Marie.l6        1.047e-01  7.175e-01   0.146  0.88412    
## Detroit.l6         -1.766e-01  8.738e-02  -2.021  0.04428 *  
## Port_Huron.l6       2.534e-01  2.793e-01   0.907  0.36513    
## B_Niagara.l6       -5.525e-02  1.139e-01  -0.485  0.62798    
## Alexandria.l6      -1.152e+00  5.689e-01  -2.024  0.04397 *  
## Ogdensburg.l6       1.399e-01  1.311e+00   0.107  0.91509    
## Massena.l6          3.439e-01  5.086e-01   0.676  0.49964    
## Champlain_Point.l6  1.240e-01  4.236e-01   0.293  0.76989    
## sd1                 1.368e+05  2.996e+04   4.567 7.71e-06 ***
## sd2                -2.886e+04  3.721e+04  -0.776  0.43863    
## sd3                 6.231e+04  3.320e+04   1.877  0.06169 .  
## sd4                 6.747e+04  3.917e+04   1.722  0.08622 .  
## sd5                 6.707e+04  3.309e+04   2.027  0.04373 *  
## sd6                 5.538e+04  3.887e+04   1.425  0.15550    
## sd7                 1.224e+05  3.734e+04   3.278  0.00119 ** 
## sd8                 2.604e+04  3.198e+04   0.814  0.41617    
## sd9                 8.616e+04  3.485e+04   2.472  0.01408 *  
## sd10                1.237e+05  4.176e+04   2.962  0.00334 ** 
## sd11                3.660e+03  4.017e+04   0.091  0.92749    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 
## Residual standard error: 60290 on 255 degrees of freedom
## Multiple R-Squared: 0.554,   Adjusted R-squared: 0.4507 
## F-statistic: 5.368 on 59 and 255 DF,  p-value: < 2.2e-16 
## 
## 
## Estimation results for equation Port_Huron: 
## =========================================== 
## Port_Huron = S_S_Marie.l1 + Detroit.l1 + Port_Huron.l1 + B_Niagara.l1 + Alexandria.l1 + Ogdensburg.l1 + Massena.l1 + Champlain_Point.l1 + S_S_Marie.l2 + Detroit.l2 + Port_Huron.l2 + B_Niagara.l2 + Alexandria.l2 + Ogdensburg.l2 + Massena.l2 + Champlain_Point.l2 + S_S_Marie.l3 + Detroit.l3 + Port_Huron.l3 + B_Niagara.l3 + Alexandria.l3 + Ogdensburg.l3 + Massena.l3 + Champlain_Point.l3 + S_S_Marie.l4 + Detroit.l4 + Port_Huron.l4 + B_Niagara.l4 + Alexandria.l4 + Ogdensburg.l4 + Massena.l4 + Champlain_Point.l4 + S_S_Marie.l5 + Detroit.l5 + Port_Huron.l5 + B_Niagara.l5 + Alexandria.l5 + Ogdensburg.l5 + Massena.l5 + Champlain_Point.l5 + S_S_Marie.l6 + Detroit.l6 + Port_Huron.l6 + B_Niagara.l6 + Alexandria.l6 + Ogdensburg.l6 + Massena.l6 + Champlain_Point.l6 + sd1 + sd2 + sd3 + sd4 + sd5 + sd6 + sd7 + sd8 + sd9 + sd10 + sd11 
## 
##                      Estimate Std. Error t value Pr(>|t|)    
## S_S_Marie.l1       -1.711e-01  2.945e-01  -0.581 0.561818    
## Detroit.l1          3.870e-02  3.304e-02   1.171 0.242554    
## Port_Huron.l1      -5.741e-01  1.092e-01  -5.257 3.10e-07 ***
## B_Niagara.l1        1.337e-02  4.583e-02   0.292 0.770707    
## Alexandria.l1      -2.175e-01  2.254e-01  -0.965 0.335586    
## Ogdensburg.l1       8.182e-01  5.147e-01   1.589 0.113189    
## Massena.l1          3.562e-01  2.072e-01   1.719 0.086758 .  
## Champlain_Point.l1 -2.106e-01  1.635e-01  -1.288 0.198825    
## S_S_Marie.l2       -4.672e-03  3.336e-01  -0.014 0.988836    
## Detroit.l2          7.574e-02  3.753e-02   2.018 0.044598 *  
## Port_Huron.l2      -4.291e-01  1.279e-01  -3.356 0.000912 ***
## B_Niagara.l2       -5.783e-02  5.533e-02  -1.045 0.296901    
## Alexandria.l2      -4.935e-01  2.765e-01  -1.785 0.075518 .  
## Ogdensburg.l2       5.799e-01  6.311e-01   0.919 0.359050    
## Massena.l2          2.577e-01  2.412e-01   1.068 0.286472    
## Champlain_Point.l2 -2.335e-01  1.946e-01  -1.200 0.231308    
## S_S_Marie.l3       -4.313e-01  3.500e-01  -1.232 0.219040    
## Detroit.l3          5.480e-02  3.948e-02   1.388 0.166399    
## Port_Huron.l3      -3.109e-01  1.401e-01  -2.219 0.027375 *  
## B_Niagara.l3       -4.171e-02  6.243e-02  -0.668 0.504686    
## Alexandria.l3      -7.388e-01  3.025e-01  -2.442 0.015287 *  
## Ogdensburg.l3       1.014e+00  7.001e-01   1.449 0.148596    
## Massena.l3          2.889e-01  2.661e-01   1.086 0.278666    
## Champlain_Point.l3  1.277e-01  2.066e-01   0.618 0.536971    
## S_S_Marie.l4       -1.093e-01  3.440e-01  -0.318 0.750988    
## Detroit.l4          6.500e-02  3.958e-02   1.642 0.101781    
## Port_Huron.l4      -2.009e-01  1.395e-01  -1.440 0.151034    
## B_Niagara.l4       -8.945e-02  6.244e-02  -1.433 0.153192    
## Alexandria.l4      -6.776e-01  3.054e-01  -2.219 0.027371 *  
## Ogdensburg.l4       1.266e+00  6.971e-01   1.816 0.070505 .  
## Massena.l4          2.121e-01  2.641e-01   0.803 0.422629    
## Champlain_Point.l4  5.193e-03  2.105e-01   0.025 0.980339    
## S_S_Marie.l5       -3.264e-01  3.239e-01  -1.008 0.314502    
## Detroit.l5          1.464e-01  3.729e-02   3.925 0.000112 ***
## Port_Huron.l5      -6.907e-02  1.305e-01  -0.529 0.597038    
## B_Niagara.l5       -9.172e-02  5.562e-02  -1.649 0.100383    
## Alexandria.l5      -2.209e-01  2.792e-01  -0.791 0.429629    
## Ogdensburg.l5       9.459e-02  6.323e-01   0.150 0.881199    
## Massena.l5          1.583e-01  2.377e-01   0.666 0.505906    
## Champlain_Point.l5  1.858e-01  2.014e-01   0.923 0.357041    
## S_S_Marie.l6       -3.169e-01  2.858e-01  -1.109 0.268625    
## Detroit.l6          6.501e-03  3.481e-02   0.187 0.852026    
## Port_Huron.l6      -1.139e-01  1.113e-01  -1.024 0.307037    
## B_Niagara.l6       -2.009e-02  4.537e-02  -0.443 0.658283    
## Alexandria.l6      -3.858e-01  2.267e-01  -1.702 0.089907 .  
## Ogdensburg.l6       3.426e-01  5.224e-01   0.656 0.512567    
## Massena.l6          2.915e-02  2.026e-01   0.144 0.885733    
## Champlain_Point.l6  2.103e-01  1.688e-01   1.246 0.213813    
## sd1                 8.787e+04  1.194e+04   7.361 2.51e-12 ***
## sd2                -2.464e+04  1.482e+04  -1.662 0.097676 .  
## sd3                 1.835e+03  1.323e+04   0.139 0.889799    
## sd4                -1.840e+03  1.561e+04  -0.118 0.906221    
## sd5                 7.858e+04  1.318e+04   5.961 8.32e-09 ***
## sd6                 1.576e+04  1.549e+04   1.017 0.309982    
## sd7                 5.543e+04  1.488e+04   3.726 0.000239 ***
## sd8                 2.324e+04  1.274e+04   1.824 0.069294 .  
## sd9                 2.268e+04  1.389e+04   1.634 0.103581    
## sd10                4.606e+04  1.664e+04   2.768 0.006048 ** 
## sd11                1.892e+04  1.601e+04   1.182 0.238212    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 
## Residual standard error: 24020 on 255 degrees of freedom
## Multiple R-Squared: 0.8272,  Adjusted R-squared: 0.7872 
## F-statistic: 20.69 on 59 and 255 DF,  p-value: < 2.2e-16 
## 
## 
## Estimation results for equation B_Niagara: 
## ========================================== 
## B_Niagara = S_S_Marie.l1 + Detroit.l1 + Port_Huron.l1 + B_Niagara.l1 + Alexandria.l1 + Ogdensburg.l1 + Massena.l1 + Champlain_Point.l1 + S_S_Marie.l2 + Detroit.l2 + Port_Huron.l2 + B_Niagara.l2 + Alexandria.l2 + Ogdensburg.l2 + Massena.l2 + Champlain_Point.l2 + S_S_Marie.l3 + Detroit.l3 + Port_Huron.l3 + B_Niagara.l3 + Alexandria.l3 + Ogdensburg.l3 + Massena.l3 + Champlain_Point.l3 + S_S_Marie.l4 + Detroit.l4 + Port_Huron.l4 + B_Niagara.l4 + Alexandria.l4 + Ogdensburg.l4 + Massena.l4 + Champlain_Point.l4 + S_S_Marie.l5 + Detroit.l5 + Port_Huron.l5 + B_Niagara.l5 + Alexandria.l5 + Ogdensburg.l5 + Massena.l5 + Champlain_Point.l5 + S_S_Marie.l6 + Detroit.l6 + Port_Huron.l6 + B_Niagara.l6 + Alexandria.l6 + Ogdensburg.l6 + Massena.l6 + Champlain_Point.l6 + sd1 + sd2 + sd3 + sd4 + sd5 + sd6 + sd7 + sd8 + sd9 + sd10 + sd11 
## 
##                      Estimate Std. Error t value Pr(>|t|)    
## S_S_Marie.l1       -1.873e-01  9.007e-01  -0.208 0.835452    
## Detroit.l1          2.498e-01  1.011e-01   2.472 0.014087 *  
## Port_Huron.l1       1.410e-01  3.340e-01   0.422 0.673334    
## B_Niagara.l1       -7.735e-01  1.401e-01  -5.519 8.38e-08 ***
## Alexandria.l1      -1.005e+00  6.894e-01  -1.458 0.145958    
## Ogdensburg.l1       2.802e+00  1.574e+00   1.780 0.076301 .  
## Massena.l1          9.115e-01  6.337e-01   1.439 0.151513    
## Champlain_Point.l1 -2.019e-01  4.999e-01  -0.404 0.686591    
## S_S_Marie.l2        2.012e-01  1.020e+00   0.197 0.843804    
## Detroit.l2          3.022e-01  1.148e-01   2.633 0.008976 ** 
## Port_Huron.l2       4.977e-01  3.910e-01   1.273 0.204278    
## B_Niagara.l2       -9.207e-01  1.692e-01  -5.441 1.24e-07 ***
## Alexandria.l2      -1.305e+00  8.457e-01  -1.543 0.124058    
## Ogdensburg.l2       9.411e-01  1.930e+00   0.488 0.626260    
## Massena.l2          3.420e-01  7.377e-01   0.464 0.643326    
## Champlain_Point.l2 -1.425e-01  5.952e-01  -0.239 0.811020    
## S_S_Marie.l3       -2.208e-01  1.070e+00  -0.206 0.836719    
## Detroit.l3          3.286e-01  1.208e-01   2.721 0.006961 ** 
## Port_Huron.l3       3.017e-02  4.286e-01   0.070 0.943931    
## B_Niagara.l3       -9.127e-01  1.909e-01  -4.780 2.96e-06 ***
## Alexandria.l3      -2.104e+00  9.253e-01  -2.274 0.023795 *  
## Ogdensburg.l3       1.774e+00  2.141e+00   0.828 0.408211    
## Massena.l3          9.686e-01  8.139e-01   1.190 0.235127    
## Champlain_Point.l3  1.070e+00  6.318e-01   1.693 0.091608 .  
## S_S_Marie.l4       -7.080e-02  1.052e+00  -0.067 0.946402    
## Detroit.l4          2.723e-01  1.211e-01   2.249 0.025367 *  
## Port_Huron.l4      -1.841e-01  4.266e-01  -0.432 0.666461    
## B_Niagara.l4       -6.221e-01  1.910e-01  -3.258 0.001275 ** 
## Alexandria.l4      -2.094e+00  9.339e-01  -2.242 0.025796 *  
## Ogdensburg.l4       3.473e+00  2.132e+00   1.629 0.104570    
## Massena.l4          1.106e+00  8.077e-01   1.370 0.171974    
## Champlain_Point.l4  4.397e-01  6.438e-01   0.683 0.495236    
## S_S_Marie.l5       -1.291e+00  9.905e-01  -1.304 0.193464    
## Detroit.l5          5.182e-01  1.140e-01   4.544 8.53e-06 ***
## Port_Huron.l5       4.629e-01  3.991e-01   1.160 0.247141    
## B_Niagara.l5       -4.732e-01  1.701e-01  -2.782 0.005808 ** 
## Alexandria.l5      -9.534e-01  8.539e-01  -1.116 0.265284    
## Ogdensburg.l5      -9.197e-01  1.934e+00  -0.476 0.634785    
## Massena.l5          7.553e-01  7.268e-01   1.039 0.299742    
## Champlain_Point.l5  7.288e-01  6.160e-01   1.183 0.237822    
## S_S_Marie.l6       -4.508e-01  8.742e-01  -0.516 0.606518    
## Detroit.l6          2.039e-01  1.065e-01   1.915 0.056587 .  
## Port_Huron.l6      -1.648e-01  3.403e-01  -0.484 0.628590    
## B_Niagara.l6       -2.336e-01  1.388e-01  -1.684 0.093416 .  
## Alexandria.l6      -1.516e+00  6.932e-01  -2.187 0.029663 *  
## Ogdensburg.l6       1.994e+00  1.598e+00   1.248 0.213231    
## Massena.l6          5.727e-01  6.198e-01   0.924 0.356322    
## Champlain_Point.l6  2.918e-01  5.161e-01   0.565 0.572324    
## sd1                 4.078e+05  3.651e+04  11.170  < 2e-16 ***
## sd2                 4.334e+03  4.534e+04   0.096 0.923913    
## sd3                 4.110e+04  4.046e+04   1.016 0.310609    
## sd4                 7.051e+04  4.773e+04   1.477 0.140864    
## sd5                 3.353e+05  4.032e+04   8.317 5.46e-15 ***
## sd6                 6.499e+04  4.737e+04   1.372 0.171242    
## sd7                 1.964e+05  4.550e+04   4.317 2.26e-05 ***
## sd8                 1.289e+05  3.896e+04   3.309 0.001072 ** 
## sd9                 6.128e+04  4.247e+04   1.443 0.150250    
## sd10                1.731e+05  5.088e+04   3.402 0.000778 ***
## sd11                1.730e+05  4.895e+04   3.533 0.000487 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 
## Residual standard error: 73460 on 255 degrees of freedom
## Multiple R-Squared: 0.9023,  Adjusted R-squared: 0.8797 
## F-statistic: 39.93 on 59 and 255 DF,  p-value: < 2.2e-16 
## 
## 
## Estimation results for equation Alexandria: 
## =========================================== 
## Alexandria = S_S_Marie.l1 + Detroit.l1 + Port_Huron.l1 + B_Niagara.l1 + Alexandria.l1 + Ogdensburg.l1 + Massena.l1 + Champlain_Point.l1 + S_S_Marie.l2 + Detroit.l2 + Port_Huron.l2 + B_Niagara.l2 + Alexandria.l2 + Ogdensburg.l2 + Massena.l2 + Champlain_Point.l2 + S_S_Marie.l3 + Detroit.l3 + Port_Huron.l3 + B_Niagara.l3 + Alexandria.l3 + Ogdensburg.l3 + Massena.l3 + Champlain_Point.l3 + S_S_Marie.l4 + Detroit.l4 + Port_Huron.l4 + B_Niagara.l4 + Alexandria.l4 + Ogdensburg.l4 + Massena.l4 + Champlain_Point.l4 + S_S_Marie.l5 + Detroit.l5 + Port_Huron.l5 + B_Niagara.l5 + Alexandria.l5 + Ogdensburg.l5 + Massena.l5 + Champlain_Point.l5 + S_S_Marie.l6 + Detroit.l6 + Port_Huron.l6 + B_Niagara.l6 + Alexandria.l6 + Ogdensburg.l6 + Massena.l6 + Champlain_Point.l6 + sd1 + sd2 + sd3 + sd4 + sd5 + sd6 + sd7 + sd8 + sd9 + sd10 + sd11 
## 
##                      Estimate Std. Error t value Pr(>|t|)    
## S_S_Marie.l1       -7.252e-02  1.327e-01  -0.546 0.585242    
## Detroit.l1          3.945e-02  1.489e-02   2.649 0.008572 ** 
## Port_Huron.l1       1.819e-02  4.921e-02   0.370 0.712036    
## B_Niagara.l1        2.843e-03  2.065e-02   0.138 0.890604    
## Alexandria.l1      -8.790e-01  1.016e-01  -8.653 5.76e-16 ***
## Ogdensburg.l1       4.109e-01  2.320e-01   1.772 0.077672 .  
## Massena.l1          1.213e-01  9.337e-02   1.300 0.194885    
## Champlain_Point.l1 -9.833e-02  7.366e-02  -1.335 0.183082    
## S_S_Marie.l2       -2.174e-01  1.503e-01  -1.446 0.149401    
## Detroit.l2          4.793e-02  1.691e-02   2.835 0.004955 ** 
## Port_Huron.l2       5.851e-02  5.762e-02   1.015 0.310842    
## B_Niagara.l2       -1.479e-02  2.493e-02  -0.593 0.553570    
## Alexandria.l2      -8.070e-01  1.246e-01  -6.476 4.82e-10 ***
## Ogdensburg.l2       2.240e-01  2.844e-01   0.788 0.431530    
## Massena.l2          8.963e-02  1.087e-01   0.825 0.410387    
## Champlain_Point.l2 -1.000e-01  8.770e-02  -1.140 0.255166    
## S_S_Marie.l3       -2.165e-01  1.577e-01  -1.373 0.171113    
## Detroit.l3          4.365e-02  1.779e-02   2.453 0.014829 *  
## Port_Huron.l3       7.665e-02  6.315e-02   1.214 0.225913    
## B_Niagara.l3       -5.647e-02  2.813e-02  -2.007 0.045790 *  
## Alexandria.l3      -7.991e-01  1.363e-01  -5.862 1.41e-08 ***
## Ogdensburg.l3       4.115e-01  3.155e-01   1.304 0.193329    
## Massena.l3          1.395e-01  1.199e-01   1.163 0.245784    
## Champlain_Point.l3  1.029e-01  9.309e-02   1.106 0.269836    
## S_S_Marie.l4       -1.825e-01  1.550e-01  -1.178 0.240091    
## Detroit.l4          3.479e-02  1.784e-02   1.950 0.052223 .  
## Port_Huron.l4       5.278e-02  6.286e-02   0.840 0.401938    
## B_Niagara.l4       -4.338e-02  2.814e-02  -1.542 0.124349    
## Alexandria.l4      -5.651e-01  1.376e-01  -4.107 5.41e-05 ***
## Ogdensburg.l4       3.785e-01  3.141e-01   1.205 0.229309    
## Massena.l4          2.998e-02  1.190e-01   0.252 0.801344    
## Champlain_Point.l4  7.224e-02  9.486e-02   0.762 0.447046    
## S_S_Marie.l5       -2.276e-01  1.459e-01  -1.560 0.120043    
## Detroit.l5          4.577e-02  1.680e-02   2.724 0.006897 ** 
## Port_Huron.l5       1.781e-01  5.880e-02   3.029 0.002702 ** 
## B_Niagara.l5       -3.989e-02  2.507e-02  -1.591 0.112752    
## Alexandria.l5      -3.535e-01  1.258e-01  -2.810 0.005344 ** 
## Ogdensburg.l5       3.971e-02  2.849e-01   0.139 0.889261    
## Massena.l5          5.285e-02  1.071e-01   0.494 0.622062    
## Champlain_Point.l5  5.270e-02  9.076e-02   0.581 0.561989    
## S_S_Marie.l6       -1.692e-01  1.288e-01  -1.314 0.190155    
## Detroit.l6          2.319e-02  1.569e-02   1.478 0.140519    
## Port_Huron.l6       3.197e-02  5.014e-02   0.638 0.524335    
## B_Niagara.l6       -2.169e-02  2.044e-02  -1.061 0.289668    
## Alexandria.l6      -3.073e-01  1.021e-01  -3.009 0.002888 ** 
## Ogdensburg.l6       4.322e-01  2.354e-01   1.836 0.067543 .  
## Massena.l6          4.305e-02  9.132e-02   0.471 0.637701    
## Champlain_Point.l6  1.635e-02  7.605e-02   0.215 0.829961    
## sd1                 5.927e+04  5.379e+03  11.019  < 2e-16 ***
## sd2                 2.943e+03  6.680e+03   0.441 0.659902    
## sd3                 6.459e+03  5.961e+03   1.084 0.279578    
## sd4                 1.059e+04  7.033e+03   1.506 0.133331    
## sd5                 5.290e+04  5.941e+03   8.905  < 2e-16 ***
## sd6                 1.266e+04  6.979e+03   1.814 0.070834 .  
## sd7                 2.894e+04  6.703e+03   4.317 2.26e-05 ***
## sd8                 1.594e+04  5.741e+03   2.777 0.005897 ** 
## sd9                 1.325e+04  6.257e+03   2.117 0.035191 *  
## sd10                2.436e+04  7.497e+03   3.249 0.001315 ** 
## sd11                2.560e+04  7.212e+03   3.550 0.000459 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 
## Residual standard error: 10820 on 255 degrees of freedom
## Multiple R-Squared: 0.9045,  Adjusted R-squared: 0.8824 
## F-statistic: 40.94 on 59 and 255 DF,  p-value: < 2.2e-16 
## 
## 
## Estimation results for equation Ogdensburg: 
## =========================================== 
## Ogdensburg = S_S_Marie.l1 + Detroit.l1 + Port_Huron.l1 + B_Niagara.l1 + Alexandria.l1 + Ogdensburg.l1 + Massena.l1 + Champlain_Point.l1 + S_S_Marie.l2 + Detroit.l2 + Port_Huron.l2 + B_Niagara.l2 + Alexandria.l2 + Ogdensburg.l2 + Massena.l2 + Champlain_Point.l2 + S_S_Marie.l3 + Detroit.l3 + Port_Huron.l3 + B_Niagara.l3 + Alexandria.l3 + Ogdensburg.l3 + Massena.l3 + Champlain_Point.l3 + S_S_Marie.l4 + Detroit.l4 + Port_Huron.l4 + B_Niagara.l4 + Alexandria.l4 + Ogdensburg.l4 + Massena.l4 + Champlain_Point.l4 + S_S_Marie.l5 + Detroit.l5 + Port_Huron.l5 + B_Niagara.l5 + Alexandria.l5 + Ogdensburg.l5 + Massena.l5 + Champlain_Point.l5 + S_S_Marie.l6 + Detroit.l6 + Port_Huron.l6 + B_Niagara.l6 + Alexandria.l6 + Ogdensburg.l6 + Massena.l6 + Champlain_Point.l6 + sd1 + sd2 + sd3 + sd4 + sd5 + sd6 + sd7 + sd8 + sd9 + sd10 + sd11 
## 
##                      Estimate Std. Error t value Pr(>|t|)    
## S_S_Marie.l1       -3.271e-02  5.083e-02  -0.643  0.52052    
## Detroit.l1          3.804e-03  5.703e-03   0.667  0.50538    
## Port_Huron.l1       2.680e-02  1.885e-02   1.422  0.15634    
## B_Niagara.l1        3.670e-03  7.909e-03   0.464  0.64302    
## Alexandria.l1      -5.414e-02  3.891e-02  -1.391  0.16533    
## Ogdensburg.l1      -6.701e-01  8.884e-02  -7.542 8.14e-13 ***
## Massena.l1         -6.729e-03  3.576e-02  -0.188  0.85089    
## Champlain_Point.l1  6.412e-03  2.821e-02   0.227  0.82040    
## S_S_Marie.l2        1.442e-02  5.758e-02   0.250  0.80242    
## Detroit.l2          7.999e-03  6.477e-03   1.235  0.21796    
## Port_Huron.l2       3.122e-02  2.207e-02   1.415  0.15835    
## B_Niagara.l2        2.108e-03  9.549e-03   0.221  0.82547    
## Alexandria.l2      -6.027e-02  4.773e-02  -1.263  0.20781    
## Ogdensburg.l2      -6.615e-01  1.089e-01  -6.073 4.54e-09 ***
## Massena.l2          1.066e-02  4.163e-02   0.256  0.79812    
## Champlain_Point.l2 -6.816e-02  3.359e-02  -2.029  0.04349 *  
## S_S_Marie.l3        3.301e-02  6.041e-02   0.546  0.58526    
## Detroit.l3          1.276e-02  6.815e-03   1.872  0.06236 .  
## Port_Huron.l3       4.381e-02  2.419e-02   1.811  0.07126 .  
## B_Niagara.l3       -1.018e-02  1.078e-02  -0.945  0.34549    
## Alexandria.l3      -8.770e-02  5.222e-02  -1.679  0.09430 .  
## Ogdensburg.l3      -6.688e-01  1.208e-01  -5.535 7.72e-08 ***
## Massena.l3         -6.504e-03  4.593e-02  -0.142  0.88751    
## Champlain_Point.l3  1.230e-02  3.566e-02   0.345  0.73033    
## S_S_Marie.l4       -6.064e-04  5.938e-02  -0.010  0.99186    
## Detroit.l4          1.007e-02  6.832e-03   1.475  0.14156    
## Port_Huron.l4       1.131e-02  2.408e-02   0.470  0.63889    
## B_Niagara.l4       -2.552e-03  1.078e-02  -0.237  0.81297    
## Alexandria.l4      -8.894e-02  5.271e-02  -1.687  0.09274 .  
## Ogdensburg.l4      -3.533e-01  1.203e-01  -2.936  0.00363 ** 
## Massena.l4         -2.568e-03  4.559e-02  -0.056  0.95512    
## Champlain_Point.l4  1.452e-02  3.633e-02   0.400  0.68973    
## S_S_Marie.l5       -5.238e-02  5.590e-02  -0.937  0.34958    
## Detroit.l5          1.748e-02  6.436e-03   2.716  0.00706 ** 
## Port_Huron.l5       1.870e-03  2.252e-02   0.083  0.93390    
## B_Niagara.l5        4.916e-03  9.600e-03   0.512  0.60902    
## Alexandria.l5      -7.450e-02  4.819e-02  -1.546  0.12336    
## Ogdensburg.l5      -2.877e-01  1.091e-01  -2.636  0.00890 ** 
## Massena.l5          6.123e-04  4.102e-02   0.015  0.98810    
## Champlain_Point.l5  4.308e-02  3.476e-02   1.239  0.21634    
## S_S_Marie.l6       -9.921e-03  4.933e-02  -0.201  0.84079    
## Detroit.l6          6.199e-03  6.009e-03   1.032  0.30325    
## Port_Huron.l6      -1.003e-02  1.920e-02  -0.522  0.60185    
## B_Niagara.l6       -2.757e-03  7.830e-03  -0.352  0.72506    
## Alexandria.l6      -2.460e-02  3.912e-02  -0.629  0.53002    
## Ogdensburg.l6      -1.378e-01  9.017e-02  -1.528  0.12769    
## Massena.l6          3.611e-02  3.498e-02   1.033  0.30281    
## Champlain_Point.l6  1.200e-02  2.913e-02   0.412  0.68075    
## sd1                 1.162e+04  2.060e+03   5.640 4.51e-08 ***
## sd2                -1.744e+03  2.559e+03  -0.682  0.49617    
## sd3                 1.270e+03  2.283e+03   0.556  0.57858    
## sd4                -4.923e+02  2.694e+03  -0.183  0.85512    
## sd5                 1.091e+04  2.275e+03   4.797 2.75e-06 ***
## sd6                 3.927e+02  2.673e+03   0.147  0.88333    
## sd7                 6.471e+03  2.568e+03   2.520  0.01233 *  
## sd8                 4.406e+03  2.199e+03   2.004  0.04613 *  
## sd9                -3.331e+01  2.397e+03  -0.014  0.98892    
## sd10                3.689e+03  2.872e+03   1.285  0.20006    
## sd11                5.387e+03  2.762e+03   1.950  0.05225 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 
## Residual standard error: 4146 on 255 degrees of freedom
## Multiple R-Squared: 0.7873,  Adjusted R-squared: 0.738 
## F-statistic: 15.99 on 59 and 255 DF,  p-value: < 2.2e-16 
## 
## 
## Estimation results for equation Massena: 
## ======================================== 
## Massena = S_S_Marie.l1 + Detroit.l1 + Port_Huron.l1 + B_Niagara.l1 + Alexandria.l1 + Ogdensburg.l1 + Massena.l1 + Champlain_Point.l1 + S_S_Marie.l2 + Detroit.l2 + Port_Huron.l2 + B_Niagara.l2 + Alexandria.l2 + Ogdensburg.l2 + Massena.l2 + Champlain_Point.l2 + S_S_Marie.l3 + Detroit.l3 + Port_Huron.l3 + B_Niagara.l3 + Alexandria.l3 + Ogdensburg.l3 + Massena.l3 + Champlain_Point.l3 + S_S_Marie.l4 + Detroit.l4 + Port_Huron.l4 + B_Niagara.l4 + Alexandria.l4 + Ogdensburg.l4 + Massena.l4 + Champlain_Point.l4 + S_S_Marie.l5 + Detroit.l5 + Port_Huron.l5 + B_Niagara.l5 + Alexandria.l5 + Ogdensburg.l5 + Massena.l5 + Champlain_Point.l5 + S_S_Marie.l6 + Detroit.l6 + Port_Huron.l6 + B_Niagara.l6 + Alexandria.l6 + Ogdensburg.l6 + Massena.l6 + Champlain_Point.l6 + sd1 + sd2 + sd3 + sd4 + sd5 + sd6 + sd7 + sd8 + sd9 + sd10 + sd11 
## 
##                      Estimate Std. Error t value Pr(>|t|)    
## S_S_Marie.l1        5.905e-03  1.044e-01   0.057 0.954961    
## Detroit.l1          5.720e-03  1.172e-02   0.488 0.625901    
## Port_Huron.l1       1.850e-02  3.873e-02   0.478 0.633326    
## B_Niagara.l1        2.813e-02  1.625e-02   1.731 0.084630 .  
## Alexandria.l1      -1.152e-01  7.995e-02  -1.441 0.150911    
## Ogdensburg.l1       1.821e-01  1.826e-01   0.998 0.319428    
## Massena.l1         -5.562e-01  7.348e-02  -7.569 6.86e-13 ***
## Champlain_Point.l1 -1.241e-01  5.797e-02  -2.141 0.033262 *  
## S_S_Marie.l2       -5.001e-02  1.183e-01  -0.423 0.672842    
## Detroit.l2          2.957e-02  1.331e-02   2.222 0.027146 *  
## Port_Huron.l2       4.633e-02  4.535e-02   1.022 0.307861    
## B_Niagara.l2       -1.492e-02  1.962e-02  -0.761 0.447611    
## Alexandria.l2       4.154e-02  9.807e-02   0.424 0.672241    
## Ogdensburg.l2       8.696e-02  2.238e-01   0.389 0.697934    
## Massena.l2         -5.968e-01  8.555e-02  -6.976 2.60e-11 ***
## Champlain_Point.l2 -1.191e-01  6.902e-02  -1.725 0.085748 .  
## S_S_Marie.l3       -9.438e-02  1.241e-01  -0.760 0.447752    
## Detroit.l3          1.555e-02  1.400e-02   1.110 0.267960    
## Port_Huron.l3      -2.756e-02  4.970e-02  -0.555 0.579706    
## B_Niagara.l3       -1.307e-02  2.214e-02  -0.590 0.555438    
## Alexandria.l3      -3.065e-02  1.073e-01  -0.286 0.775370    
## Ogdensburg.l3       6.297e-02  2.483e-01   0.254 0.800016    
## Massena.l3         -4.092e-01  9.438e-02  -4.336 2.09e-05 ***
## Champlain_Point.l3  6.431e-02  7.326e-02   0.878 0.380861    
## S_S_Marie.l4       -8.992e-02  1.220e-01  -0.737 0.461761    
## Detroit.l4          1.239e-02  1.404e-02   0.883 0.378238    
## Port_Huron.l4      -3.725e-02  4.947e-02  -0.753 0.452161    
## B_Niagara.l4       -8.908e-04  2.214e-02  -0.040 0.967941    
## Alexandria.l4      -7.376e-02  1.083e-01  -0.681 0.496418    
## Ogdensburg.l4       1.882e-01  2.472e-01   0.761 0.447155    
## Massena.l4         -2.916e-01  9.367e-02  -3.113 0.002064 ** 
## Champlain_Point.l4 -1.190e-03  7.465e-02  -0.016 0.987300    
## S_S_Marie.l5        5.712e-02  1.149e-01   0.497 0.619374    
## Detroit.l5          2.991e-02  1.322e-02   2.262 0.024533 *  
## Port_Huron.l5       5.023e-02  4.628e-02   1.085 0.278744    
## B_Niagara.l5       -1.080e-02  1.973e-02  -0.547 0.584573    
## Alexandria.l5      -8.068e-02  9.902e-02  -0.815 0.415952    
## Ogdensburg.l5      -2.299e-01  2.242e-01  -1.025 0.306166    
## Massena.l5         -2.018e-01  8.428e-02  -2.394 0.017382 *  
## Champlain_Point.l5 -2.061e-02  7.143e-02  -0.289 0.773143    
## S_S_Marie.l6        5.377e-02  1.014e-01   0.530 0.596285    
## Detroit.l6         -1.524e-02  1.235e-02  -1.234 0.218188    
## Port_Huron.l6       5.313e-02  3.946e-02   1.346 0.179358    
## B_Niagara.l6       -3.771e-03  1.609e-02  -0.234 0.814876    
## Alexandria.l6      -1.217e-01  8.038e-02  -1.514 0.131389    
## Ogdensburg.l6       1.708e-01  1.853e-01   0.922 0.357573    
## Massena.l6         -8.831e-02  7.187e-02  -1.229 0.220281    
## Champlain_Point.l6 -5.804e-02  5.985e-02  -0.970 0.333053    
## sd1                 2.353e+04  4.233e+03   5.558 6.86e-08 ***
## sd2                 2.986e+03  5.257e+03   0.568 0.570586    
## sd3                 4.672e+03  4.691e+03   0.996 0.320228    
## sd4                 3.798e+03  5.535e+03   0.686 0.493207    
## sd5                 2.531e+04  4.675e+03   5.413 1.43e-07 ***
## sd6                 8.523e+03  5.493e+03   1.552 0.121952    
## sd7                 1.909e+04  5.276e+03   3.618 0.000357 ***
## sd8                 1.095e+04  4.518e+03   2.424 0.016031 *  
## sd9                 8.379e+03  4.924e+03   1.702 0.090061 .  
## sd10                2.394e+04  5.900e+03   4.057 6.61e-05 ***
## sd11                9.089e+03  5.676e+03   1.601 0.110581    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 
## Residual standard error: 8519 on 255 degrees of freedom
## Multiple R-Squared: 0.7254,  Adjusted R-squared: 0.6619 
## F-statistic: 11.42 on 59 and 255 DF,  p-value: < 2.2e-16 
## 
## 
## Estimation results for equation Champlain_Point: 
## ================================================ 
## Champlain_Point = S_S_Marie.l1 + Detroit.l1 + Port_Huron.l1 + B_Niagara.l1 + Alexandria.l1 + Ogdensburg.l1 + Massena.l1 + Champlain_Point.l1 + S_S_Marie.l2 + Detroit.l2 + Port_Huron.l2 + B_Niagara.l2 + Alexandria.l2 + Ogdensburg.l2 + Massena.l2 + Champlain_Point.l2 + S_S_Marie.l3 + Detroit.l3 + Port_Huron.l3 + B_Niagara.l3 + Alexandria.l3 + Ogdensburg.l3 + Massena.l3 + Champlain_Point.l3 + S_S_Marie.l4 + Detroit.l4 + Port_Huron.l4 + B_Niagara.l4 + Alexandria.l4 + Ogdensburg.l4 + Massena.l4 + Champlain_Point.l4 + S_S_Marie.l5 + Detroit.l5 + Port_Huron.l5 + B_Niagara.l5 + Alexandria.l5 + Ogdensburg.l5 + Massena.l5 + Champlain_Point.l5 + S_S_Marie.l6 + Detroit.l6 + Port_Huron.l6 + B_Niagara.l6 + Alexandria.l6 + Ogdensburg.l6 + Massena.l6 + Champlain_Point.l6 + sd1 + sd2 + sd3 + sd4 + sd5 + sd6 + sd7 + sd8 + sd9 + sd10 + sd11 
## 
##                      Estimate Std. Error t value Pr(>|t|)    
## S_S_Marie.l1       -3.730e-02  1.914e-01  -0.195 0.845616    
## Detroit.l1          5.040e-02  2.147e-02   2.347 0.019684 *  
## Port_Huron.l1       4.802e-02  7.097e-02   0.677 0.499284    
## B_Niagara.l1       -1.759e-02  2.978e-02  -0.591 0.555158    
## Alexandria.l1      -7.324e-02  1.465e-01  -0.500 0.617516    
## Ogdensburg.l1       1.566e-02  3.345e-01   0.047 0.962698    
## Massena.l1          1.462e-01  1.346e-01   1.086 0.278700    
## Champlain_Point.l1 -7.432e-01  1.062e-01  -6.997 2.30e-11 ***
## S_S_Marie.l2       -1.413e-01  2.168e-01  -0.652 0.515041    
## Detroit.l2          5.962e-02  2.438e-02   2.445 0.015163 *  
## Port_Huron.l2       1.129e-01  8.309e-02   1.359 0.175419    
## B_Niagara.l2       -4.556e-02  3.595e-02  -1.267 0.206192    
## Alexandria.l2      -2.157e-01  1.797e-01  -1.201 0.231033    
## Ogdensburg.l2      -1.937e-01  4.101e-01  -0.472 0.637158    
## Massena.l2          6.260e-02  1.567e-01   0.399 0.689944    
## Champlain_Point.l2 -7.003e-01  1.265e-01  -5.538 7.62e-08 ***
## S_S_Marie.l3       -5.393e-02  2.274e-01  -0.237 0.812771    
## Detroit.l3          8.323e-02  2.566e-02   3.244 0.001337 ** 
## Port_Huron.l3       8.437e-02  9.106e-02   0.927 0.355038    
## B_Niagara.l3       -8.168e-02  4.057e-02  -2.013 0.045132 *  
## Alexandria.l3      -3.590e-01  1.966e-01  -1.826 0.069042 .  
## Ogdensburg.l3      -1.789e-01  4.549e-01  -0.393 0.694423    
## Massena.l3          1.220e-01  1.729e-01   0.705 0.481249    
## Champlain_Point.l3 -3.641e-01  1.342e-01  -2.713 0.007132 ** 
## S_S_Marie.l4        1.204e-01  2.235e-01   0.538 0.590741    
## Detroit.l4          5.815e-02  2.572e-02   2.261 0.024628 *  
## Port_Huron.l4       9.490e-02  9.065e-02   1.047 0.296155    
## B_Niagara.l4       -8.667e-02  4.057e-02  -2.136 0.033614 *  
## Alexandria.l4      -2.374e-01  1.984e-01  -1.196 0.232741    
## Ogdensburg.l4       8.503e-03  4.530e-01   0.019 0.985039    
## Massena.l4          8.573e-02  1.716e-01   0.500 0.617854    
## Champlain_Point.l4 -3.202e-01  1.368e-01  -2.341 0.020026 *  
## S_S_Marie.l5       -1.628e-02  2.105e-01  -0.077 0.938398    
## Detroit.l5          9.495e-02  2.423e-02   3.919 0.000114 ***
## Port_Huron.l5       1.256e-01  8.480e-02   1.481 0.139796    
## B_Niagara.l5       -7.391e-02  3.614e-02  -2.045 0.041908 *  
## Alexandria.l5      -1.427e-01  1.814e-01  -0.786 0.432377    
## Ogdensburg.l5       3.745e-02  4.109e-01   0.091 0.927454    
## Massena.l5          1.064e-01  1.544e-01   0.689 0.491306    
## Champlain_Point.l5 -1.673e-01  1.309e-01  -1.278 0.202330    
## S_S_Marie.l6       -1.817e-01  1.857e-01  -0.978 0.328860    
## Detroit.l6          5.712e-02  2.262e-02   2.525 0.012182 *  
## Port_Huron.l6      -3.439e-02  7.230e-02  -0.476 0.634717    
## B_Niagara.l6       -3.296e-02  2.948e-02  -1.118 0.264645    
## Alexandria.l6      -8.702e-02  1.473e-01  -0.591 0.555144    
## Ogdensburg.l6       4.505e-02  3.395e-01   0.133 0.894518    
## Massena.l6          2.499e-02  1.317e-01   0.190 0.849643    
## Champlain_Point.l6 -5.048e-02  1.097e-01  -0.460 0.645687    
## sd1                 8.427e+04  7.757e+03  10.865  < 2e-16 ***
## sd2                -1.094e+03  9.633e+03  -0.114 0.909666    
## sd3                 1.237e+04  8.596e+03   1.439 0.151409    
## sd4                 1.446e+04  1.014e+04   1.426 0.155230    
## sd5                 7.954e+04  8.567e+03   9.284  < 2e-16 ***
## sd6                 1.572e+04  1.006e+04   1.562 0.119627    
## sd7                 3.272e+04  9.667e+03   3.385 0.000823 ***
## sd8                 2.605e+04  8.279e+03   3.147 0.001846 ** 
## sd9                 9.115e+03  9.023e+03   1.010 0.313333    
## sd10                3.352e+04  1.081e+04   3.100 0.002150 ** 
## sd11                3.167e+04  1.040e+04   3.045 0.002572 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 
## Residual standard error: 15610 on 255 degrees of freedom
## Multiple R-Squared: 0.902,   Adjusted R-squared: 0.8793 
## F-statistic: 39.78 on 59 and 255 DF,  p-value: < 2.2e-16 
## 
## 
## 
## Covariance matrix of residuals:
##                 S_S_Marie   Detroit Port_Huron B_Niagara Alexandria Ogdensburg
## S_S_Marie        73242283 2.775e+08  1.372e+08 4.949e+08   60274661   22184127
## Detroit         277475716 3.628e+09  9.115e+08 2.707e+09  318203627  106120836
## Port_Huron      137197183 9.115e+08  5.765e+08 1.368e+09  172115518   58829189
## B_Niagara       494889515 2.707e+09  1.368e+09 5.382e+09  621768243  186936601
## Alexandria       60274661 3.182e+08  1.721e+08 6.218e+08  117040519   25738662
## Ogdensburg       22184127 1.061e+08  5.883e+07 1.869e+08   25738662   17179219
## Massena          32633829 2.209e+08  9.801e+07 3.012e+08   40893775   10370716
## Champlain_Point  88352541 4.718e+08  2.682e+08 8.688e+08  114418973   43178371
##                   Massena Champlain_Point
## S_S_Marie        32633829        88352541
## Detroit         220904248       471827833
## Port_Huron       98007913       268178013
## B_Niagara       301203218       868764447
## Alexandria       40893775       114418973
## Ogdensburg       10370716        43178371
## Massena          72478594        58835239
## Champlain_Point  58835239       243642990
## 
## Correlation matrix of residuals:
##                 S_S_Marie Detroit Port_Huron B_Niagara Alexandria Ogdensburg
## S_S_Marie          1.0000  0.5383     0.6677    0.7883     0.6510     0.6254
## Detroit            0.5383  1.0000     0.6302    0.6125     0.4883     0.4251
## Port_Huron         0.6677  0.6302     1.0000    0.7768     0.6626     0.5911
## B_Niagara          0.7883  0.6125     0.7768    1.0000     0.7834     0.6148
## Alexandria         0.6510  0.4883     0.6626    0.7834     1.0000     0.5740
## Ogdensburg         0.6254  0.4251     0.5911    0.6148     0.5740     1.0000
## Massena            0.4479  0.4308     0.4795    0.4823     0.4440     0.2939
## Champlain_Point    0.6614  0.5018     0.7156    0.7587     0.6776     0.6674
##                 Massena Champlain_Point
## S_S_Marie        0.4479          0.6614
## Detroit          0.4308          0.5018
## Port_Huron       0.4795          0.7156
## B_Niagara        0.4823          0.7587
## Alexandria       0.4440          0.6776
## Ogdensburg       0.2939          0.6674
## Massena          1.0000          0.4427
## Champlain_Point  0.4427          1.0000
selected_order <- var.a$selection$ic[1, "lag"]

# Print the selected lag order
print(selected_order)
## NULL
VARselect(stnry_ts, lag.max=8, type="none", season = 12)$selection
## AIC(n)  HQ(n)  SC(n) FPE(n) 
##      6      3      1      6
var <- vars::VAR(stnry_ts, p=2, season = 12, type = "none", ic = c("SC"))
summary(var)
## 
## VAR Estimation Results:
## ========================= 
## Endogenous variables: S_S_Marie, Detroit, Port_Huron, B_Niagara, Alexandria, Ogdensburg, Massena, Champlain_Point 
## Deterministic variables: none 
## Sample size: 318 
## Log Likelihood: -27538.11 
## Roots of the characteristic polynomial:
## 0.7037 0.7037 0.6716 0.6716 0.6145 0.6145 0.5986 0.5986 0.5798 0.5798 0.4839 0.4839 0.4462 0.4462 0.3807 0.3807
## Call:
## vars::VAR(y = stnry_ts, p = 2, type = "none", season = 12L, ic = c("SC"))
## 
## 
## Estimation results for equation S_S_Marie: 
## ========================================== 
## S_S_Marie = S_S_Marie.l1 + Detroit.l1 + Port_Huron.l1 + B_Niagara.l1 + Alexandria.l1 + Ogdensburg.l1 + Massena.l1 + Champlain_Point.l1 + S_S_Marie.l2 + Detroit.l2 + Port_Huron.l2 + B_Niagara.l2 + Alexandria.l2 + Ogdensburg.l2 + Massena.l2 + Champlain_Point.l2 + sd1 + sd2 + sd3 + sd4 + sd5 + sd6 + sd7 + sd8 + sd9 + sd10 + sd11 
## 
##                      Estimate Std. Error t value Pr(>|t|)    
## S_S_Marie.l1       -5.197e-01  1.045e-01  -4.974 1.13e-06 ***
## Detroit.l1          2.014e-02  1.229e-02   1.638   0.1024    
## Port_Huron.l1      -1.690e-02  3.867e-02  -0.437   0.6624    
## B_Niagara.l1       -7.944e-03  1.464e-02  -0.543   0.5877    
## Alexandria.l1       4.776e-02  7.806e-02   0.612   0.5412    
## Ogdensburg.l1       2.901e-01  1.727e-01   1.680   0.0941 .  
## Massena.l1          7.303e-02  7.050e-02   1.036   0.3011    
## Champlain_Point.l1 -1.111e-02  5.720e-02  -0.194   0.8462    
## S_S_Marie.l2       -2.348e-01  1.035e-01  -2.268   0.0241 *  
## Detroit.l2         -5.614e-03  1.237e-02  -0.454   0.6502    
## Port_Huron.l2       7.187e-02  3.818e-02   1.883   0.0608 .  
## B_Niagara.l2       -1.413e-02  1.490e-02  -0.948   0.3437    
## Alexandria.l2       3.433e-02  7.915e-02   0.434   0.6648    
## Ogdensburg.l2       7.881e-03  1.701e-01   0.046   0.9631    
## Massena.l2         -1.208e-02  6.877e-02  -0.176   0.8606    
## Champlain_Point.l2 -9.436e-02  5.783e-02  -1.632   0.1038    
## sd1                 3.280e+04  3.278e+03  10.006  < 2e-16 ***
## sd2                -1.550e+04  3.790e+03  -4.090 5.59e-05 ***
## sd3                 6.309e+03  3.900e+03   1.618   0.1068    
## sd4                 1.069e+03  4.258e+03   0.251   0.8019    
## sd5                 3.947e+04  3.105e+03  12.711  < 2e-16 ***
## sd6                 9.188e+03  3.801e+03   2.417   0.0163 *  
## sd7                 8.780e+03  3.898e+03   2.253   0.0250 *  
## sd8                 6.256e+03  3.996e+03   1.565   0.1186    
## sd9                -1.481e+03  3.320e+03  -0.446   0.6558    
## sd10                2.138e+04  3.283e+03   6.510 3.29e-10 ***
## sd11                1.863e+04  3.171e+03   5.873 1.17e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 
## Residual standard error: 9929 on 291 degrees of freedom
## Multiple R-Squared: 0.7805,  Adjusted R-squared: 0.7601 
## F-statistic: 38.32 on 27 and 291 DF,  p-value: < 2.2e-16 
## 
## 
## Estimation results for equation Detroit: 
## ======================================== 
## Detroit = S_S_Marie.l1 + Detroit.l1 + Port_Huron.l1 + B_Niagara.l1 + Alexandria.l1 + Ogdensburg.l1 + Massena.l1 + Champlain_Point.l1 + S_S_Marie.l2 + Detroit.l2 + Port_Huron.l2 + B_Niagara.l2 + Alexandria.l2 + Ogdensburg.l2 + Massena.l2 + Champlain_Point.l2 + sd1 + sd2 + sd3 + sd4 + sd5 + sd6 + sd7 + sd8 + sd9 + sd10 + sd11 
## 
##                      Estimate Std. Error t value Pr(>|t|)    
## S_S_Marie.l1        6.280e-01  6.768e-01   0.928  0.35417    
## Detroit.l1         -3.595e-01  7.962e-02  -4.515 9.21e-06 ***
## Port_Huron.l1       4.796e-02  2.505e-01   0.191  0.84830    
## B_Niagara.l1       -1.173e-01  9.481e-02  -1.238  0.21684    
## Alexandria.l1       2.860e-02  5.057e-01   0.057  0.95493    
## Ogdensburg.l1       3.400e-01  1.119e+00   0.304  0.76137    
## Massena.l1          1.910e-01  4.567e-01   0.418  0.67609    
## Champlain_Point.l1  1.490e-01  3.705e-01   0.402  0.68790    
## S_S_Marie.l2        2.043e-01  6.706e-01   0.305  0.76083    
## Detroit.l2         -2.220e-01  8.010e-02  -2.772  0.00593 ** 
## Port_Huron.l2       1.622e-01  2.473e-01   0.656  0.51229    
## B_Niagara.l2       -6.244e-02  9.654e-02  -0.647  0.51825    
## Alexandria.l2       1.237e-02  5.127e-01   0.024  0.98076    
## Ogdensburg.l2      -8.984e-01  1.102e+00  -0.815  0.41549    
## Massena.l2          1.047e-01  4.454e-01   0.235  0.81429    
## Champlain_Point.l2 -1.072e-01  3.745e-01  -0.286  0.77488    
## sd1                 9.517e+04  2.123e+04   4.482 1.06e-05 ***
## sd2                -7.417e+04  2.455e+04  -3.021  0.00274 ** 
## sd3                 3.333e+04  2.526e+04   1.319  0.18810    
## sd4                 2.481e+04  2.758e+04   0.900  0.36896    
## sd5                 6.394e+04  2.011e+04   3.179  0.00164 ** 
## sd6                 7.328e+04  2.462e+04   2.977  0.00316 ** 
## sd7                 6.247e+04  2.525e+04   2.474  0.01392 *  
## sd8                 2.044e+04  2.589e+04   0.790  0.43046    
## sd9                -6.945e+03  2.151e+04  -0.323  0.74695    
## sd10                6.843e+04  2.127e+04   3.217  0.00144 ** 
## sd11                1.184e+04  2.054e+04   0.576  0.56481    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 
## Residual standard error: 64320 on 291 degrees of freedom
## Multiple R-Squared: 0.4226,  Adjusted R-squared: 0.369 
## F-statistic: 7.887 on 27 and 291 DF,  p-value: < 2.2e-16 
## 
## 
## Estimation results for equation Port_Huron: 
## =========================================== 
## Port_Huron = S_S_Marie.l1 + Detroit.l1 + Port_Huron.l1 + B_Niagara.l1 + Alexandria.l1 + Ogdensburg.l1 + Massena.l1 + Champlain_Point.l1 + S_S_Marie.l2 + Detroit.l2 + Port_Huron.l2 + B_Niagara.l2 + Alexandria.l2 + Ogdensburg.l2 + Massena.l2 + Champlain_Point.l2 + sd1 + sd2 + sd3 + sd4 + sd5 + sd6 + sd7 + sd8 + sd9 + sd10 + sd11 
## 
##                      Estimate Std. Error t value Pr(>|t|)    
## S_S_Marie.l1       -8.881e-02  2.788e-01  -0.319   0.7503    
## Detroit.l1          3.670e-02  3.280e-02   1.119   0.2641    
## Port_Huron.l1      -4.995e-01  1.032e-01  -4.841 2.10e-06 ***
## B_Niagara.l1        1.081e-02  3.905e-02   0.277   0.7820    
## Alexandria.l1       3.787e-02  2.083e-01   0.182   0.8558    
## Ogdensburg.l1       4.800e-01  4.607e-01   1.042   0.2983    
## Massena.l1          1.857e-01  1.881e-01   0.988   0.3242    
## Champlain_Point.l1 -1.498e-01  1.526e-01  -0.981   0.3272    
## S_S_Marie.l2        2.562e-01  2.762e-01   0.927   0.3545    
## Detroit.l2          2.565e-03  3.299e-02   0.078   0.9381    
## Port_Huron.l2      -1.750e-01  1.018e-01  -1.718   0.0868 .  
## B_Niagara.l2       -3.538e-02  3.976e-02  -0.890   0.3743    
## Alexandria.l2      -4.062e-02  2.112e-01  -0.192   0.8476    
## Ogdensburg.l2      -1.040e-02  4.538e-01  -0.023   0.9817    
## Massena.l2          6.288e-02  1.835e-01   0.343   0.7320    
## Champlain_Point.l2 -2.767e-01  1.543e-01  -1.793   0.0739 .  
## sd1                 7.782e+04  8.745e+03   8.898  < 2e-16 ***
## sd2                -5.151e+04  1.011e+04  -5.095 6.28e-07 ***
## sd3                -4.840e+03  1.041e+04  -0.465   0.6422    
## sd4                -1.759e+04  1.136e+04  -1.549   0.1226    
## sd5                 9.130e+04  8.284e+03  11.021  < 2e-16 ***
## sd6                 4.232e+04  1.014e+04   4.173 3.97e-05 ***
## sd7                 2.027e+04  1.040e+04   1.949   0.0522 .  
## sd8                 1.262e+04  1.066e+04   1.184   0.2374    
## sd9                -7.441e+03  8.858e+03  -0.840   0.4016    
## sd10                4.042e+04  8.760e+03   4.614 5.94e-06 ***
## sd11                1.811e+04  8.461e+03   2.141   0.0331 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 
## Residual standard error: 26490 on 291 degrees of freedom
## Multiple R-Squared: 0.7626,  Adjusted R-squared: 0.7405 
## F-statistic: 34.61 on 27 and 291 DF,  p-value: < 2.2e-16 
## 
## 
## Estimation results for equation B_Niagara: 
## ========================================== 
## B_Niagara = S_S_Marie.l1 + Detroit.l1 + Port_Huron.l1 + B_Niagara.l1 + Alexandria.l1 + Ogdensburg.l1 + Massena.l1 + Champlain_Point.l1 + S_S_Marie.l2 + Detroit.l2 + Port_Huron.l2 + B_Niagara.l2 + Alexandria.l2 + Ogdensburg.l2 + Massena.l2 + Champlain_Point.l2 + sd1 + sd2 + sd3 + sd4 + sd5 + sd6 + sd7 + sd8 + sd9 + sd10 + sd11 
## 
##                      Estimate Std. Error t value Pr(>|t|)    
## S_S_Marie.l1       -4.521e-01  9.527e-01  -0.475 0.635480    
## Detroit.l1          1.992e-01  1.121e-01   1.777 0.076568 .  
## Port_Huron.l1       6.945e-02  3.526e-01   0.197 0.843992    
## B_Niagara.l1       -4.970e-01  1.335e-01  -3.724 0.000235 ***
## Alexandria.l1      -5.166e-01  7.118e-01  -0.726 0.468560    
## Ogdensburg.l1       1.035e+00  1.575e+00   0.657 0.511523    
## Massena.l1          5.466e-01  6.428e-01   0.850 0.395807    
## Champlain_Point.l1  2.322e-01  5.216e-01   0.445 0.656481    
## S_S_Marie.l2        6.776e-01  9.440e-01   0.718 0.473491    
## Detroit.l2          5.071e-02  1.128e-01   0.450 0.653233    
## Port_Huron.l2       6.840e-01  3.481e-01   1.965 0.050351 .  
## B_Niagara.l2       -4.415e-01  1.359e-01  -3.249 0.001295 ** 
## Alexandria.l2      -7.419e-02  7.217e-01  -0.103 0.918198    
## Ogdensburg.l2      -8.016e-01  1.551e+00  -0.517 0.605666    
## Massena.l2         -1.021e+00  6.270e-01  -1.628 0.104565    
## Champlain_Point.l2 -4.169e-01  5.272e-01  -0.791 0.429735    
## sd1                 3.390e+05  2.989e+04  11.344  < 2e-16 ***
## sd2                -1.968e+05  3.455e+04  -5.696 3.00e-08 ***
## sd3                -2.074e+04  3.556e+04  -0.583 0.560279    
## sd4                -7.119e+04  3.882e+04  -1.834 0.067712 .  
## sd5                 3.920e+05  2.831e+04  13.846  < 2e-16 ***
## sd6                 7.409e+04  3.466e+04   2.138 0.033373 *  
## sd7                 1.556e+04  3.554e+04   0.438 0.661835    
## sd8                 1.921e+04  3.644e+04   0.527 0.598533    
## sd9                -4.438e+04  3.027e+04  -1.466 0.143756    
## sd10                1.468e+05  2.994e+04   4.904 1.57e-06 ***
## sd11                8.752e+04  2.892e+04   3.027 0.002693 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 
## Residual standard error: 90540 on 291 degrees of freedom
## Multiple R-Squared: 0.835,   Adjusted R-squared: 0.8196 
## F-statistic: 54.53 on 27 and 291 DF,  p-value: < 2.2e-16 
## 
## 
## Estimation results for equation Alexandria: 
## =========================================== 
## Alexandria = S_S_Marie.l1 + Detroit.l1 + Port_Huron.l1 + B_Niagara.l1 + Alexandria.l1 + Ogdensburg.l1 + Massena.l1 + Champlain_Point.l1 + S_S_Marie.l2 + Detroit.l2 + Port_Huron.l2 + B_Niagara.l2 + Alexandria.l2 + Ogdensburg.l2 + Massena.l2 + Champlain_Point.l2 + sd1 + sd2 + sd3 + sd4 + sd5 + sd6 + sd7 + sd8 + sd9 + sd10 + sd11 
## 
##                      Estimate Std. Error t value Pr(>|t|)    
## S_S_Marie.l1       -5.090e-02  1.380e-01  -0.369 0.712589    
## Detroit.l1          3.644e-02  1.624e-02   2.244 0.025608 *  
## Port_Huron.l1      -1.067e-02  5.109e-02  -0.209 0.834677    
## B_Niagara.l1        1.489e-02  1.934e-02   0.770 0.442107    
## Alexandria.l1      -6.303e-01  1.031e-01  -6.111 3.17e-09 ***
## Ogdensburg.l1       2.136e-01  2.281e-01   0.936 0.349930    
## Massena.l1          4.194e-02  9.315e-02   0.450 0.652869    
## Champlain_Point.l1 -3.882e-02  7.558e-02  -0.514 0.607896    
## S_S_Marie.l2       -6.416e-02  1.368e-01  -0.469 0.639410    
## Detroit.l2          1.737e-02  1.634e-02   1.063 0.288565    
## Port_Huron.l2       3.517e-02  5.044e-02   0.697 0.486116    
## B_Niagara.l2        7.853e-03  1.969e-02   0.399 0.690304    
## Alexandria.l2      -2.986e-01  1.046e-01  -2.856 0.004604 ** 
## Ogdensburg.l2      -3.167e-02  2.247e-01  -0.141 0.888030    
## Massena.l2         -6.896e-02  9.086e-02  -0.759 0.448449    
## Champlain_Point.l2 -1.312e-01  7.640e-02  -1.717 0.087062 .  
## sd1                 5.190e+04  4.331e+03  11.984  < 2e-16 ***
## sd2                -2.432e+04  5.007e+03  -4.857 1.95e-06 ***
## sd3                 5.803e+02  5.153e+03   0.113 0.910410    
## sd4                -8.198e+03  5.625e+03  -1.457 0.146078    
## sd5                 6.237e+04  4.102e+03  15.204  < 2e-16 ***
## sd6                 1.398e+04  5.022e+03   2.784 0.005717 ** 
## sd7                 1.285e+03  5.150e+03   0.250 0.803117    
## sd8                 1.310e+03  5.280e+03   0.248 0.804275    
## sd9                 3.891e+03  4.386e+03   0.887 0.375735    
## sd10                2.551e+04  4.338e+03   5.881 1.12e-08 ***
## sd11                1.398e+04  4.190e+03   3.336 0.000961 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 
## Residual standard error: 13120 on 291 degrees of freedom
## Multiple R-Squared: 0.8435,  Adjusted R-squared: 0.829 
## F-statistic:  58.1 on 27 and 291 DF,  p-value: < 2.2e-16 
## 
## 
## Estimation results for equation Ogdensburg: 
## =========================================== 
## Ogdensburg = S_S_Marie.l1 + Detroit.l1 + Port_Huron.l1 + B_Niagara.l1 + Alexandria.l1 + Ogdensburg.l1 + Massena.l1 + Champlain_Point.l1 + S_S_Marie.l2 + Detroit.l2 + Port_Huron.l2 + B_Niagara.l2 + Alexandria.l2 + Ogdensburg.l2 + Massena.l2 + Champlain_Point.l2 + sd1 + sd2 + sd3 + sd4 + sd5 + sd6 + sd7 + sd8 + sd9 + sd10 + sd11 
## 
##                      Estimate Std. Error t value Pr(>|t|)    
## S_S_Marie.l1       -6.859e-02  4.859e-02  -1.412 0.159128    
## Detroit.l1          3.304e-03  5.717e-03   0.578 0.563738    
## Port_Huron.l1       4.363e-03  1.798e-02   0.243 0.808457    
## B_Niagara.l1        8.812e-03  6.807e-03   1.295 0.196484    
## Alexandria.l1      -2.857e-02  3.630e-02  -0.787 0.431973    
## Ogdensburg.l1      -4.377e-01  8.030e-02  -5.451 1.07e-07 ***
## Massena.l1         -1.858e-03  3.279e-02  -0.057 0.954844    
## Champlain_Point.l1  2.008e-02  2.660e-02   0.755 0.451042    
## S_S_Marie.l2       -2.813e-02  4.815e-02  -0.584 0.559557    
## Detroit.l2          2.498e-04  5.751e-03   0.043 0.965378    
## Port_Huron.l2       5.464e-03  1.775e-02   0.308 0.758459    
## B_Niagara.l2        5.316e-03  6.931e-03   0.767 0.443712    
## Alexandria.l2       5.381e-03  3.681e-02   0.146 0.883866    
## Ogdensburg.l2      -2.852e-01  7.910e-02  -3.606 0.000366 ***
## Massena.l2          1.115e-02  3.198e-02   0.349 0.727605    
## Champlain_Point.l2 -5.637e-02  2.689e-02  -2.096 0.036938 *  
## sd1                 1.171e+04  1.524e+03   7.679 2.45e-13 ***
## sd2                -6.633e+03  1.762e+03  -3.763 0.000203 ***
## sd3                 2.153e+03  1.814e+03   1.187 0.236086    
## sd4                -2.597e+03  1.980e+03  -1.312 0.190592    
## sd5                 1.484e+04  1.444e+03  10.273  < 2e-16 ***
## sd6                 2.908e+03  1.768e+03   1.645 0.101035    
## sd7                 2.148e+03  1.813e+03   1.185 0.236967    
## sd8                 2.010e+03  1.859e+03   1.081 0.280471    
## sd9                 1.747e+02  1.544e+03   0.113 0.909996    
## sd10                7.042e+03  1.527e+03   4.612 5.99e-06 ***
## sd11                3.875e+03  1.475e+03   2.627 0.009061 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 
## Residual standard error: 4618 on 291 degrees of freedom
## Multiple R-Squared: 0.7048,  Adjusted R-squared: 0.6774 
## F-statistic: 25.73 on 27 and 291 DF,  p-value: < 2.2e-16 
## 
## 
## Estimation results for equation Massena: 
## ======================================== 
## Massena = S_S_Marie.l1 + Detroit.l1 + Port_Huron.l1 + B_Niagara.l1 + Alexandria.l1 + Ogdensburg.l1 + Massena.l1 + Champlain_Point.l1 + S_S_Marie.l2 + Detroit.l2 + Port_Huron.l2 + B_Niagara.l2 + Alexandria.l2 + Ogdensburg.l2 + Massena.l2 + Champlain_Point.l2 + sd1 + sd2 + sd3 + sd4 + sd5 + sd6 + sd7 + sd8 + sd9 + sd10 + sd11 
## 
##                      Estimate Std. Error t value Pr(>|t|)    
## S_S_Marie.l1       -5.701e-03  9.792e-02  -0.058   0.9536    
## Detroit.l1          5.101e-03  1.152e-02   0.443   0.6583    
## Port_Huron.l1       1.752e-03  3.624e-02   0.048   0.9615    
## B_Niagara.l1        3.003e-02  1.372e-02   2.189   0.0294 *  
## Alexandria.l1      -9.495e-02  7.316e-02  -1.298   0.1954    
## Ogdensburg.l1       2.417e-02  1.618e-01   0.149   0.8814    
## Massena.l1         -4.036e-01  6.607e-02  -6.108 3.22e-09 ***
## Champlain_Point.l1 -8.234e-02  5.361e-02  -1.536   0.1256    
## S_S_Marie.l2        5.973e-03  9.703e-02   0.062   0.9510    
## Detroit.l2          1.773e-02  1.159e-02   1.530   0.1271    
## Port_Huron.l2       5.462e-02  3.578e-02   1.527   0.1279    
## B_Niagara.l2       -1.429e-02  1.397e-02  -1.023   0.3072    
## Alexandria.l2       6.830e-02  7.418e-02   0.921   0.3579    
## Ogdensburg.l2       6.485e-02  1.594e-01   0.407   0.6845    
## Massena.l2         -4.029e-01  6.445e-02  -6.252 1.44e-09 ***
## Champlain_Point.l2 -1.016e-01  5.419e-02  -1.874   0.0619 .  
## sd1                 1.753e+04  3.072e+03   5.707 2.83e-08 ***
## sd2                -7.656e+03  3.551e+03  -2.156   0.0319 *  
## sd3                 4.805e+02  3.655e+03   0.131   0.8955    
## sd4                -2.446e+03  3.990e+03  -0.613   0.5403    
## sd5                 2.442e+04  2.910e+03   8.392 2.11e-15 ***
## sd6                 8.249e+03  3.562e+03   2.316   0.0213 *  
## sd7                 7.897e+03  3.653e+03   2.162   0.0314 *  
## sd8                 5.021e+03  3.745e+03   1.341   0.1811    
## sd9                -3.815e+03  3.111e+03  -1.226   0.2211    
## sd10                1.413e+04  3.077e+03   4.591 6.59e-06 ***
## sd11                2.986e+03  2.972e+03   1.005   0.3160    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 
## Residual standard error: 9305 on 291 degrees of freedom
## Multiple R-Squared: 0.6324,  Adjusted R-squared: 0.5983 
## F-statistic: 18.54 on 27 and 291 DF,  p-value: < 2.2e-16 
## 
## 
## Estimation results for equation Champlain_Point: 
## ================================================ 
## Champlain_Point = S_S_Marie.l1 + Detroit.l1 + Port_Huron.l1 + B_Niagara.l1 + Alexandria.l1 + Ogdensburg.l1 + Massena.l1 + Champlain_Point.l1 + S_S_Marie.l2 + Detroit.l2 + Port_Huron.l2 + B_Niagara.l2 + Alexandria.l2 + Ogdensburg.l2 + Massena.l2 + Champlain_Point.l2 + sd1 + sd2 + sd3 + sd4 + sd5 + sd6 + sd7 + sd8 + sd9 + sd10 + sd11 
## 
##                      Estimate Std. Error t value Pr(>|t|)    
## S_S_Marie.l1       -9.240e-02  1.866e-01  -0.495  0.62096    
## Detroit.l1          3.952e-02  2.196e-02   1.800  0.07293 .  
## Port_Huron.l1       1.159e-02  6.908e-02   0.168  0.86688    
## B_Niagara.l1        6.589e-03  2.615e-02   0.252  0.80122    
## Alexandria.l1      -3.530e-03  1.395e-01  -0.025  0.97982    
## Ogdensburg.l1       1.579e-01  3.085e-01   0.512  0.60924    
## Massena.l1          7.874e-02  1.259e-01   0.625  0.53232    
## Champlain_Point.l1 -5.268e-01  1.022e-01  -5.155 4.70e-07 ***
## S_S_Marie.l2       -7.096e-02  1.850e-01  -0.384  0.70150    
## Detroit.l2         -3.005e-03  2.209e-02  -0.136  0.89188    
## Port_Huron.l2       8.893e-02  6.820e-02   1.304  0.19323    
## B_Niagara.l2       -1.557e-02  2.662e-02  -0.585  0.55917    
## Alexandria.l2       5.524e-02  1.414e-01   0.391  0.69634    
## Ogdensburg.l2      -1.282e-01  3.039e-01  -0.422  0.67340    
## Massena.l2         -7.616e-02  1.228e-01  -0.620  0.53576    
## Champlain_Point.l2 -4.302e-01  1.033e-01  -4.164 4.12e-05 ***
## sd1                 7.085e+04  5.855e+03  12.099  < 2e-16 ***
## sd2                -4.108e+04  6.770e+03  -6.068 4.03e-09 ***
## sd3                 3.188e+03  6.967e+03   0.458  0.64760    
## sd4                -1.427e+04  7.606e+03  -1.876  0.06162 .  
## sd5                 8.857e+04  5.547e+03  15.968  < 2e-16 ***
## sd6                 1.508e+04  6.790e+03   2.220  0.02718 *  
## sd7                -3.988e+03  6.963e+03  -0.573  0.56731    
## sd8                 1.731e+03  7.139e+03   0.242  0.80859    
## sd9                -6.274e+03  5.931e+03  -1.058  0.29104    
## sd10                3.483e+04  5.866e+03   5.937 8.23e-09 ***
## sd11                1.807e+04  5.665e+03   3.189  0.00158 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 
## Residual standard error: 17740 on 291 degrees of freedom
## Multiple R-Squared: 0.8586,  Adjusted R-squared: 0.8455 
## F-statistic: 65.47 on 27 and 291 DF,  p-value: < 2.2e-16 
## 
## 
## 
## Covariance matrix of residuals:
##                 S_S_Marie   Detroit Port_Huron B_Niagara Alexandria Ogdensburg
## S_S_Marie        98190338 3.504e+08  1.812e+08 7.428e+08   95236480   29015160
## Detroit         350355533 4.133e+09  1.140e+09 3.544e+09  423069383  123549616
## Port_Huron      181217837 1.140e+09  7.017e+08 1.900e+09  242475925   68027249
## B_Niagara       742758412 3.544e+09  1.900e+09 8.193e+09  991700687  256197746
## Alexandria       95236480 4.231e+08  2.425e+08 9.917e+08  172094772   35000465
## Ogdensburg       29015160 1.235e+08  6.803e+07 2.562e+08   35000465   21315968
## Massena          45281274 2.909e+08  1.286e+08 4.453e+08   59492373   13218819
## Champlain_Point 124389888 5.585e+08  3.388e+08 1.248e+09  168695814   55733975
##                   Massena Champlain_Point
## S_S_Marie        45281274       1.244e+08
## Detroit         290899826       5.585e+08
## Port_Huron      128583715       3.388e+08
## B_Niagara       445321246       1.248e+09
## Alexandria       59492373       1.687e+08
## Ogdensburg       13218819       5.573e+07
## Massena          86561809       7.278e+07
## Champlain_Point  72783064       3.146e+08
## 
## Correlation matrix of residuals:
##                 S_S_Marie Detroit Port_Huron B_Niagara Alexandria Ogdensburg
## S_S_Marie          1.0000  0.5500     0.6904    0.8281     0.7326     0.6342
## Detroit            0.5500  1.0000     0.6695    0.6090     0.5016     0.4162
## Port_Huron         0.6904  0.6695     1.0000    0.7923     0.6978     0.5562
## B_Niagara          0.8281  0.6090     0.7923    1.0000     0.8352     0.6131
## Alexandria         0.7326  0.5016     0.6978    0.8352     1.0000     0.5779
## Ogdensburg         0.6342  0.4162     0.5562    0.6131     0.5779     1.0000
## Massena            0.4912  0.4863     0.5217    0.5288     0.4874     0.3077
## Champlain_Point    0.7077  0.4898     0.7211    0.7774     0.7250     0.6806
##                 Massena Champlain_Point
## S_S_Marie        0.4912          0.7077
## Detroit          0.4863          0.4898
## Port_Huron       0.5217          0.7211
## B_Niagara        0.5288          0.7774
## Alexandria       0.4874          0.7250
## Ogdensburg       0.3077          0.6806
## Massena          1.0000          0.4410
## Champlain_Point  0.4410          1.0000
# Residual diagnostics
serial.test(var)
## 
##  Portmanteau Test (asymptotic)
## 
## data:  Residuals of VAR object var
## Chi-squared = 1582.5, df = 896, p-value < 2.2e-16
resid.var <- resid(var)
plot.ts(resid.var,ylab = 'Residuals',xlab='Year')

# ACF of residuals
acf(resid(var), lag.max = 25)

# PACF of residuals
#pacf(resid.var)
pacf(resid(var))

#selecting the variables
# Granger test for causality
#for causality function to give reliable results we need all the variables of the multivariate time series to be stationary. 
# Run the Granger causality tests for each cause variable
granger_results <- lapply(c("S_S_Marie", "Detroit", "Port_Huron", "B_Niagara", 
                             "Alexandria", "Ogdensburg", "Massena", "Champlain_Point"),
                           function(cause_var) {
                             causality(var.a, cause = cause_var)
                           })

# Extract p-values for Granger causality and Instant tests for each port
p_values <- sapply(granger_results, function(result) {
  c(Granger = result$Granger$p.value, Instant = result$Instant$p.value)
})

# Convert the p-values into a data frame for better readability
p_values_df <- as.data.frame(p_values)
colnames(p_values_df) <- c("S_S_Marie", "Detroit", "Port_Huron", "B_Niagara", 
                            "Alexandria", "Ogdensburg", "Massena", "Champlain_Point")

# Print or further analyze the results
print(t(p_values_df))
##                     Granger      Instant
## S_S_Marie       0.399929931 0.000000e+00
## Detroit         0.002567269 0.000000e+00
## Port_Huron      0.010626893 0.000000e+00
## B_Niagara       0.059133132 0.000000e+00
## Alexandria      0.317753694 0.000000e+00
## Ogdensburg      0.033622568 0.000000e+00
## Massena         0.954878920 3.701484e-13
## Champlain_Point 0.136128747 0.000000e+00
# Forecast next 5 steps
pred_var <- predict(var, n.ahead = 5, ci = 0.95)
pred_var
## $S_S_Marie
##           fcst      lower      upper       CI
## [1,] -20772.92 -40234.268 -1311.5627 19461.35
## [2,]  13959.33  -7154.523 35073.1802 21113.85
## [3,]  11920.56  -9822.885 33664.0029 21743.44
## [4,] -21830.58 -43945.540   284.3868 22114.96
## [5,]  26649.10   4490.964 48807.2387 22158.14
## 
## $Detroit
##            fcst      lower     upper       CI
## [1,] -69030.416 -195086.91  57026.07 126056.5
## [2,]  38109.268  -95731.82 171950.35 133841.1
## [3,]   -526.438 -135958.98 134906.11 135432.5
## [4,] -42202.056 -178590.67  94186.56 136388.6
## [5,]  66703.873  -69821.48 203229.22 136525.4
## 
## $Port_Huron
##            fcst     lower     upper       CI
## [1,] -29916.707 -81837.54  22004.12 51920.83
## [2,]  29912.573 -26586.91  86412.06 56499.49
## [3,]   3831.647 -53965.93  61629.23 57797.58
## [4,] -34633.737 -93753.94  24486.47 59120.20
## [5,]  71254.895  12034.92 130474.87 59219.98
## 
## $B_Niagara
##            fcst      lower     upper       CI
## [1,] -190069.68 -367515.73 -12623.62 177446.1
## [2,]  113009.35  -80055.64 306074.33 193065.0
## [3,]   67019.32 -131628.86 265667.50 198648.2
## [4,] -143410.26 -346587.43  59766.91 203177.2
## [5,]  311700.50  107992.62 515408.38 203707.9
## 
## $Alexandria
##            fcst     lower     upper       CI
## [1,] -16351.510 -42063.35  9360.334 25711.84
## [2,]  16708.205 -12191.63 45608.040 28899.84
## [3,]   2860.326 -26517.35 32238.001 29377.68
## [4,] -21273.695 -51281.03  8733.640 30007.33
## [5,]  48076.143  17933.42 78218.868 30142.73
## 
## $Ogdensburg
##           fcst       lower     upper        CI
## [1,] -7106.390 -16156.6616  1943.881  9050.271
## [2,]  4926.983  -4850.0801 14704.046  9777.063
## [3,]  2281.777  -7830.5937 12394.148 10112.371
## [4,] -6614.895 -16993.4306  3763.640 10378.536
## [5,] 10119.308   -284.3543 20522.971 10403.663
## 
## $Massena
##             fcst      lower     upper       CI
## [1,] -15421.0121 -33659.028  2817.004 18238.02
## [2,]  11672.2780  -7934.703 31279.259 19606.98
## [3,]    714.6443 -19649.571 21078.859 20364.22
## [4,] -10464.0237 -31350.605 10422.558 20886.58
## [5,]  14522.1337  -6459.135 35503.402 20981.27
## 
## $Champlain_Point
##           fcst      lower      upper       CI
## [1,] -38442.86 -73208.366  -3677.355 34765.51
## [2,]  28686.15  -9263.512  66635.818 37949.67
## [3,]  11710.02 -27729.883  51149.927 39439.91
## [4,] -33431.23 -74112.424   7249.957 40681.19
## [5,]  65000.35  24232.541 105768.150 40767.80
# plot the forecasts
par(mar=c(3,3,1,1))
plot(pred_var)

# Comparison with univariate AR models #

# time series length for out-of-sample forecasting
T0 <- 12
# length of training data set
T <- dim(multivariate_ts)[1] - T0

ts <- multivariate_ts[1:T,]
ts1 <- ts[-(1:9),1]
ts2 <- ts[-(1:9),2]

# univariate prediction

auto.arima(ts2,max.p=5,max.q=5,max.P = 0,max.Q=0,ic="aic",allowmean = TRUE)
## Series: ts2 
## ARIMA(0,1,2) 
## 
## Coefficients:
##           ma1      ma2
##       -0.5245  -0.1642
## s.e.   0.0561   0.0541
## 
## sigma^2 = 5.16e+09:  log likelihood = -3766.99
## AIC=7539.97   AICc=7540.05   BIC=7551.07
auto.arima(ts1,max.p=5,max.q=5,max.P = 0,max.Q=0,ic="aic",allowmean = TRUE)
## Series: ts1 
## ARIMA(1,1,5) with drift 
## 
## Coefficients:
##           ar1      ma1      ma2     ma3     ma4     ma5      drift
##       -0.1291  -0.8294  -0.1676  0.0153  0.1037  0.1407  -440.5315
## s.e.   0.3480   0.3430   0.3347  0.0772  0.0720  0.0891   194.4874
## 
## sigma^2 = 213330552:  log likelihood = -3288.92
## AIC=6593.85   AICc=6594.35   BIC=6623.45
#sarima(y,p=0,q=3,P=0,Q=1,S=4,d=0)

#ts1_ar <- sarima(ts1, p=1,q=1,P=0,Q=0,S=12,d=0, include.mean = TRUE, method = "ML")
#ts2_ar <- sarima(ts2, p=0,q=0,P=0,Q=0,S=12,d=0, include.mean = TRUE, method = "ML")

ts1_ar <- arima(ts1, order=c(1,0,1), include.mean = TRUE, method = "ML")
ts2_ar <- arima(ts2, order=c(0,0,0), include.mean = TRUE, method = "ML")

ts1.forecast <- predict(ts1_ar, T0)
ts2.forecast <- predict(ts2_ar, T0)

diff1 <- mean((ts1.forecast$pred[1:T0] - multivariate_ts[(T+1):(T+T0),1])^2)
diff2 <- mean((ts2.forecast$pred[1:T0] - multivariate_ts[(T+1):(T+T0),2])^2)

# Forecast error from univariate model
diff1
## [1] 280653543
diff2
## [1] 94427104252
diff1+diff2
## [1] 94707757795
# Prediction using multivariate model
var0 <- vars::VAR(ts[-(1:9),], p=2, type="both", season = 12)
pred_var0 <- predict(var0, n.ahead = T0, ci = 0.95)

# Forecast error from multivariate model
diff1_m <- mean((pred_var0$fcst$S_S_Marie[,1] - multivariate_ts[(T+1):(T+T0),1])^2)
diff2_m <- mean((pred_var0$fcst$Detroit[,1] - multivariate_ts[(T+1):(T+T0),2])^2)
diff1_m
## [1] 141455315
# Comparison with univariate ARIMA models #

# time series length for out-of-sample forecasting
T0 <- 12
# length of training data set
T <- dim(stnry_ts)[1] - T0

ts <- stnry_ts[1:T, ]
ts1 <- ts[-(1:9), 1]
ts2 <- ts[-(1:9), 2]

# Univariate ARIMA model for ts1
arima_ts1 <- auto.arima(ts1, max.p = 5, max.q = 5, max.P = 0, max.Q = 0, ic = "aic", allowmean = TRUE)
ts1_ar <- arima(ts1, order = arima_ts1$arma[c(1, 1, 5)], include.mean = TRUE, method = "ML")

# Univariate ARIMA model for ts2
arima_ts2 <- auto.arima(ts2, max.p = 5, max.q = 5, max.P = 0, max.Q = 0, ic = "aic", allowmean = TRUE)
ts2_ar <- arima(ts2, order = arima_ts2$arma[c(1, 1, 2)], include.mean = TRUE, method = "ML")

# Forecast using ARIMA models
ts1_forecast <- predict(ts1_ar, n.ahead = T0)
ts2_forecast <- predict(ts2_ar, n.ahead = T0)

# Calculate forecast errors for ARIMA models
diff1_ar <- mean((ts1_forecast$pred - stnry_ts[(T + 1):(T + T0), 1])^2)
diff2_ar <- mean((ts2_forecast$pred - stnry_ts[(T + 1):(T + T0), 2])^2)

# Multivariate VAR model
var0 <- vars::VAR(ts[-(1:9), ], p = 2, type = "both", season = 12)
pred_var0 <- predict(var0, n.ahead = T0, ci = 0.95)

# Forecast errors for VAR model
diff1_var <- mean((pred_var0$fcst$S_S_Marie[, 1] - stnry_ts[(T + 1):(T + T0), 1])^2)
diff2_var <- mean((pred_var0$fcst$Detroit[, 1] - stnry_ts[(T + 1):(T + T0), 2])^2)
diff1_var_multivariate <- mean((pred_var0$fcst$S_S_Marie[, 1] - stnry_ts[(T + 1):(T + T0), 1])^2)

# Create data frame for forecast errors
forecast_errors <- data.frame(
  Model = c(rep("ARIMA Univariate", each = 3), rep("VAR Multivariate", each = 3)),
  Series = rep(c("Yt_1", "Yt_2", "Yt_1 + Yt_2"), 2),
  Forecast_Error = c(diff1_ar, diff2_ar, diff1_ar + diff2_ar, diff1_var, diff2_var, diff1_var + diff2_var)
)

# Print forecast errors table
print(forecast_errors)
##              Model      Series Forecast_Error
## 1 ARIMA Univariate        Yt_1      262452430
## 2 ARIMA Univariate        Yt_2     7777405896
## 3 ARIMA Univariate Yt_1 + Yt_2     8039858326
## 4 VAR Multivariate        Yt_1      345681233
## 5 VAR Multivariate        Yt_2     6057116931
## 6 VAR Multivariate Yt_1 + Yt_2     6402798165